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 ⌀ |
|---|---|---|---|---|---|---|
_root_.StarAlgHom.ext_topologicalClosure [T2Space B] {S : StarSubalgebra R A}
{φ ψ : S.topologicalClosure →⋆ₐ[R] B} (hφ : Continuous φ) (hψ : Continuous ψ)
(h :
φ.comp (inclusion (le_topologicalClosure S)) = ψ.comp (inclusion (le_topologicalClosure S))) :
φ = ψ := by
rw [DFunLike.ext'_iff]
have : ... | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | _root_.StarAlgHom.ext_topologicalClosure | Continuous `StarAlgHom`s from the topological closure of a `StarSubalgebra` whose
compositions with the `StarSubalgebra.inclusion` map agree are, in fact, equal. |
_root_.StarAlgHomClass.ext_topologicalClosure [T2Space B] {F : Type*}
{S : StarSubalgebra R A} [FunLike F S.topologicalClosure B]
[AlgHomClass F R S.topologicalClosure B] [StarHomClass F S.topologicalClosure B] {φ ψ : F}
(hφ : Continuous φ) (hψ : Continuous ψ) (h : ∀ x : S,
φ (inclusion (le_topologi... | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | _root_.StarAlgHomClass.ext_topologicalClosure | null |
elemental (x : A) : StarSubalgebra R A :=
(adjoin R ({x} : Set A)).topologicalClosure | def | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | elemental | The topological closure of the star subalgebra generated by a single element. |
self_mem (x : A) : x ∈ elemental R x :=
le_topologicalClosure _ (self_mem_adjoin_singleton R x)
@[simp, aesop safe (rule_sets := [SetLike])] | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | self_mem | null |
star_self_mem (x : A) : star x ∈ elemental R x :=
star_mem <| self_mem R x | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | star_self_mem | null |
isClosedEmbedding_coe (x : A) : IsClosedEmbedding ((↑) : elemental R x → A) where
eq_induced := rfl
injective := Subtype.coe_injective
isClosed_range := by simpa using isClosed R x | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | isClosedEmbedding_coe | The `elemental` star subalgebra generated by a normal element is commutative. -/
instance [T2Space A] {x : A} [IsStarNormal x] : CommSemiring (elemental R x) :=
StarSubalgebra.commSemiringTopologicalClosure _ mul_comm
/-- The `elemental` generated by a normal element is commutative. -/
instance {R A} [CommRing R] [S... |
le_centralizer_centralizer [T2Space A] (x : A) :
elemental R x ≤ centralizer R (centralizer R {x}) :=
topologicalClosure_adjoin_le_centralizer_centralizer ..
@[elab_as_elim] | lemma | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | le_centralizer_centralizer | null |
induction_on {x y : A}
(hy : y ∈ elemental R x) {P : (u : A) → u ∈ elemental R x → Prop}
(self : P x (self_mem R x)) (star_self : P (star x) (star_self_mem R x))
(algebraMap : ∀ r, P (algebraMap R A r) (_root_.algebraMap_mem _ r))
(add : ∀ u hu v hv, P u hu → P v hv → P (u + v) (add_mem hu hv))
(mul... | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | induction_on | null |
starAlgHomClass_ext [T2Space B] {F : Type*} {a : A}
[FunLike F (elemental R a) B] [AlgHomClass F R _ B] [StarHomClass F _ B]
{φ ψ : F} (hφ : Continuous φ)
(hψ : Continuous ψ) (h : φ ⟨a, self_mem R a⟩ = ψ ⟨a, self_mem R a⟩) : φ = ψ := by
refine StarAlgHomClass.ext_topologicalClosure hφ hψ fun x => ?_
ref... | theorem | Topology | [
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/StarSubalgebra.lean | starAlgHomClass_ext | null |
@[to_additive /-- The topological support of a function is the closure of its support. i.e. the
closure of the set of all elements where the function is nonzero. -/]
mulTSupport (f : X → α) : Set X := closure (mulSupport f)
@[to_additive] | def | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | mulTSupport | The topological support of a function is the closure of its support, i.e. the closure of the
set of all elements where the function is not equal to 1. |
subset_mulTSupport (f : X → α) : mulSupport f ⊆ mulTSupport f :=
subset_closure
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | subset_mulTSupport | null |
isClosed_mulTSupport (f : X → α) : IsClosed (mulTSupport f) :=
isClosed_closure
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | isClosed_mulTSupport | null |
mulTSupport_eq_empty_iff {f : X → α} : mulTSupport f = ∅ ↔ f = 1 := by
rw [mulTSupport, closure_empty_iff, mulSupport_eq_empty_iff]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | mulTSupport_eq_empty_iff | null |
image_eq_one_of_notMem_mulTSupport {f : X → α} {x : X} (hx : x ∉ mulTSupport f) : f x = 1 :=
mulSupport_subset_iff'.mp (subset_mulTSupport f) x hx
@[deprecated (since := "2025-05-24")]
alias image_eq_zero_of_nmem_tsupport := image_eq_zero_of_notMem_tsupport
@[to_additive existing, deprecated (since := "2025-05-24")]
... | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | image_eq_one_of_notMem_mulTSupport | null |
range_subset_insert_image_mulTSupport (f : X → α) :
range f ⊆ insert 1 (f '' mulTSupport f) := by
grw [← subset_mulTSupport f]; exact range_subset_insert_image_mulSupport f
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | range_subset_insert_image_mulTSupport | null |
range_eq_image_mulTSupport_or (f : X → α) :
range f = f '' mulTSupport f ∨ range f = insert 1 (f '' mulTSupport f) :=
(wcovBy_insert _ _).eq_or_eq (image_subset_range _ _) (range_subset_insert_image_mulTSupport f) | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | range_eq_image_mulTSupport_or | null |
tsupport_mul_subset_left {α : Type*} [MulZeroClass α] {f g : X → α} :
(tsupport fun x => f x * g x) ⊆ tsupport f :=
closure_mono (support_mul_subset_left _ _) | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | tsupport_mul_subset_left | null |
tsupport_mul_subset_right {α : Type*} [MulZeroClass α] {f g : X → α} :
(tsupport fun x => f x * g x) ⊆ tsupport g :=
closure_mono (support_mul_subset_right _ _) | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | tsupport_mul_subset_right | null |
tsupport_smul_subset_left {M α} [TopologicalSpace X] [Zero M] [Zero α] [SMulWithZero M α]
(f : X → M) (g : X → α) : (tsupport fun x => f x • g x) ⊆ tsupport f :=
closure_mono <| support_smul_subset_left f g | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | tsupport_smul_subset_left | null |
tsupport_smul_subset_right {M α} [TopologicalSpace X] [Zero α] [SMulZeroClass M α]
(f : X → M) (g : X → α) : (tsupport fun x => f x • g x) ⊆ tsupport g :=
closure_mono <| support_smul_subset_right f g
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | tsupport_smul_subset_right | null |
mulTSupport_mul [TopologicalSpace X] [MulOneClass α] {f g : X → α} :
(mulTSupport fun x ↦ f x * g x) ⊆ mulTSupport f ∪ mulTSupport g :=
closure_minimal
((mulSupport_mul f g).trans (union_subset_union (subset_mulTSupport _) (subset_mulTSupport _)))
(isClosed_closure.union isClosed_closure) | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | mulTSupport_mul | null |
@[to_additive]
notMem_mulTSupport_iff_eventuallyEq : x ∉ mulTSupport f ↔ f =ᶠ[𝓝 x] 1 := by
simp_rw [mulTSupport, mem_closure_iff_nhds, not_forall, not_nonempty_iff_eq_empty, exists_prop,
← disjoint_iff_inter_eq_empty, disjoint_mulSupport_iff, eventuallyEq_iff_exists_mem]
@[deprecated (since := "2025-05-23")]
ali... | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | notMem_mulTSupport_iff_eventuallyEq | null |
continuous_of_mulTSupport [TopologicalSpace β] {f : α → β}
(hf : ∀ x ∈ mulTSupport f, ContinuousAt f x) : Continuous f :=
continuous_iff_continuousAt.2 fun x => (em _).elim (hf x) fun hx =>
(@continuousAt_const _ _ _ _ _ 1).congr (notMem_mulTSupport_iff_eventuallyEq.mp hx).symm
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | continuous_of_mulTSupport | null |
ContinuousOn.continuous_of_mulTSupport_subset [TopologicalSpace β] {f : α → β}
{s : Set α} (hs : ContinuousOn f s) (h's : IsOpen s) (h''s : mulTSupport f ⊆ s) :
Continuous f :=
continuous_of_mulTSupport fun _ hx ↦ h's.continuousOn_iff.mp hs <| h''s hx | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | ContinuousOn.continuous_of_mulTSupport_subset | null |
@[to_additive /-- A function `f` *has compact support* or is *compactly supported* if the closure of
the support of `f` is compact. In a T₂ space this is equivalent to `f` being equal to `0` outside a
compact set. -/]
HasCompactMulSupport (f : α → β) : Prop :=
IsCompact (mulTSupport f)
@[to_additive] | def | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | HasCompactMulSupport | A function `f` *has compact multiplicative support* or is *compactly supported* if the closure
of the multiplicative support of `f` is compact. In a T₂ space this is equivalent to `f` being equal
to `1` outside a compact set. |
hasCompactMulSupport_def : HasCompactMulSupport f ↔ IsCompact (closure (mulSupport f)) := by
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | hasCompactMulSupport_def | null |
exists_compact_iff_hasCompactMulSupport [R1Space α] :
(∃ K : Set α, IsCompact K ∧ ∀ x, x ∉ K → f x = 1) ↔ HasCompactMulSupport f := by
simp_rw [← notMem_mulSupport, ← mem_compl_iff, ← subset_def, compl_subset_compl,
hasCompactMulSupport_def, exists_isCompact_superset_iff] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | exists_compact_iff_hasCompactMulSupport | null |
@[to_additive]
intro [R1Space α] (hK : IsCompact K) (hfK : ∀ x, x ∉ K → f x = 1) :
HasCompactMulSupport f :=
exists_compact_iff_hasCompactMulSupport.mp ⟨K, hK, hfK⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | intro | null |
intro' (hK : IsCompact K) (h'K : IsClosed K) (hfK : ∀ x, x ∉ K → f x = 1) :
HasCompactMulSupport f := by
have : mulTSupport f ⊆ K := by
rw [← h'K.closure_eq]
apply closure_mono (mulSupport_subset_iff'.2 hfK)
exact IsCompact.of_isClosed_subset hK ( isClosed_mulTSupport f) this
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | intro' | null |
of_mulSupport_subset_isCompact [R1Space α] (hK : IsCompact K) (h : mulSupport f ⊆ K) :
HasCompactMulSupport f :=
hK.closure_of_subset h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | of_mulSupport_subset_isCompact | null |
isCompact (hf : HasCompactMulSupport f) : IsCompact (mulTSupport f) := hf
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | isCompact | null |
_root_.hasCompactMulSupport_iff_eventuallyEq :
HasCompactMulSupport f ↔ f =ᶠ[coclosedCompact α] 1 :=
mem_coclosedCompact_iff.symm
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | _root_.hasCompactMulSupport_iff_eventuallyEq | null |
_root_.isCompact_range_of_mulSupport_subset_isCompact [TopologicalSpace β]
(hf : Continuous f) (hk : IsCompact K) (h'f : mulSupport f ⊆ K) :
IsCompact (range f) := by
rcases range_eq_image_or_of_mulSupport_subset h'f with h2 | h2 <;> rw [h2]
exacts [hk.image hf, (hk.image hf).insert 1]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | _root_.isCompact_range_of_mulSupport_subset_isCompact | null |
isCompact_range [TopologicalSpace β] (h : HasCompactMulSupport f)
(hf : Continuous f) : IsCompact (range f) :=
isCompact_range_of_mulSupport_subset_isCompact hf h (subset_mulTSupport f)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | isCompact_range | null |
mono' {f' : α → γ} (hf : HasCompactMulSupport f) (hff' : mulSupport f' ⊆ mulTSupport f) :
HasCompactMulSupport f' :=
IsCompact.of_isClosed_subset hf isClosed_closure <| closure_minimal hff' isClosed_closure
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | mono' | null |
mono {f' : α → γ} (hf : HasCompactMulSupport f) (hff' : mulSupport f' ⊆ mulSupport f) :
HasCompactMulSupport f' :=
hf.mono' <| hff'.trans subset_closure
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | mono | null |
comp_left (hf : HasCompactMulSupport f) (hg : g 1 = 1) :
HasCompactMulSupport (g ∘ f) :=
hf.mono <| mulSupport_comp_subset hg f
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | comp_left | null |
_root_.hasCompactMulSupport_comp_left (hg : ∀ {x}, g x = 1 ↔ x = 1) :
HasCompactMulSupport (g ∘ f) ↔ HasCompactMulSupport f := by
simp_rw [hasCompactMulSupport_def, mulSupport_comp_eq g (@hg) f]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | _root_.hasCompactMulSupport_comp_left | null |
comp_isClosedEmbedding (hf : HasCompactMulSupport f) {g : α' → α}
(hg : IsClosedEmbedding g) : HasCompactMulSupport (f ∘ g) := by
rw [hasCompactMulSupport_def, Function.mulSupport_comp_eq_preimage]
refine IsCompact.of_isClosed_subset (hg.isCompact_preimage hf) isClosed_closure ?_
rw [hg.isEmbedding.closure_eq... | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | comp_isClosedEmbedding | null |
comp₂_left (hf : HasCompactMulSupport f)
(hf₂ : HasCompactMulSupport f₂) (hm : m 1 1 = 1) :
HasCompactMulSupport fun x => m (f x) (f₂ x) := by
rw [hasCompactMulSupport_iff_eventuallyEq] at hf hf₂ ⊢
filter_upwards [hf, hf₂] with x hx hx₂
simp_rw [hx, hx₂, Pi.one_apply, hm]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | comp₂_left | null |
isCompact_preimage [TopologicalSpace β] {K : Set β}
(h'f : HasCompactMulSupport f) (hf : Continuous f) (hk : IsClosed K) (h'k : 1 ∉ K) :
IsCompact (f ⁻¹' K) := by
apply IsCompact.of_isClosed_subset h'f (hk.preimage hf) (fun x hx ↦ ?_)
apply subset_mulTSupport
aesop
variable [T2Space α'] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | isCompact_preimage | null |
@[to_additive]
mulTSupport_extend_one_subset :
mulTSupport (g.extend f 1) ⊆ g '' mulTSupport f :=
(hf.image cont).isClosed.closure_subset_iff.mpr <|
mulSupport_extend_one_subset.trans (image_mono subset_closure)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | mulTSupport_extend_one_subset | null |
extend_one : HasCompactMulSupport (g.extend f 1) :=
HasCompactMulSupport.of_mulSupport_subset_isCompact (hf.image cont)
(subset_closure.trans <| hf.mulTSupport_extend_one_subset cont)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | extend_one | null |
mulTSupport_extend_one (inj : g.Injective) :
mulTSupport (g.extend f 1) = g '' mulTSupport f :=
(hf.mulTSupport_extend_one_subset cont).antisymm <|
(image_closure_subset_closure_image cont).trans
(closure_mono (mulSupport_extend_one inj).superset) | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | mulTSupport_extend_one | null |
@[to_additive]
continuous_extend_one [TopologicalSpace β] {U : Set α'} (hU : IsOpen U) {f : U → β}
(cont : Continuous f) (supp : HasCompactMulSupport f) :
Continuous (Subtype.val.extend f 1) :=
continuous_of_mulTSupport fun x h ↦ by
rw [show x = ↑(⟨x, Subtype.coe_image_subset _ _
(supp.mulTSupport_e... | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | continuous_extend_one | null |
@[to_additive /-- If `f` has compact support, then `f` tends to zero at infinity. -/]
is_one_at_infty {f : α → γ} [TopologicalSpace γ]
(h : HasCompactMulSupport f) : Tendsto f (cocompact α) (𝓝 1) := by
intro N hN
rw [mem_map, mem_cocompact']
refine ⟨mulTSupport f, h.isCompact, ?_⟩
rw [compl_subset_comm]
... | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | is_one_at_infty | If `f` has compact multiplicative support, then `f` tends to 1 at infinity. |
@[to_additive]
HasCompactMulSupport.of_compactSpace (f : α → γ) :
HasCompactMulSupport f :=
IsCompact.of_isClosed_subset isCompact_univ (isClosed_mulTSupport f)
(Set.subset_univ (mulTSupport f)) | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | HasCompactMulSupport.of_compactSpace | In a compact space `α`, any function has compact support. |
@[to_additive]
HasCompactMulSupport.mul (hf : HasCompactMulSupport f) (hf' : HasCompactMulSupport f') :
HasCompactMulSupport (f * f') := hf.comp₂_left hf' (mul_one 1)
@[to_additive, simp] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | HasCompactMulSupport.mul | null |
protected HasCompactMulSupport.one {α β : Type*} [TopologicalSpace α] [One β] :
HasCompactMulSupport (1 : α → β) := by
simp [HasCompactMulSupport, mulTSupport] | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | HasCompactMulSupport.one | null |
@[to_additive]
protected HasCompactMulSupport.inv {α β : Type*} [TopologicalSpace α] [DivisionMonoid β]
{f : α → β} (hf : HasCompactMulSupport f) :
HasCompactMulSupport (f⁻¹) := by
simpa only [HasCompactMulSupport, mulTSupport, mulSupport_inv] using hf
@[deprecated (since := "2025-07-31")] alias HasCompactSup... | lemma | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | HasCompactMulSupport.inv | null |
HasCompactSupport.div {α β : Type*} [TopologicalSpace α] [DivisionMonoid β]
{f f' : α → β} (hf : HasCompactMulSupport f) (hf' : HasCompactMulSupport f') :
HasCompactMulSupport (f / f') :=
div_eq_mul_inv f f' ▸ hf.mul hf'.inv | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | HasCompactSupport.div | null |
HasCompactSupport.smul_left (hf : HasCompactSupport f') : HasCompactSupport (f • f') := by
rw [hasCompactSupport_iff_eventuallyEq] at hf ⊢
exact hf.mono fun x hx => by simp_rw [Pi.smul_apply', hx, Pi.zero_apply, smul_zero] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | HasCompactSupport.smul_left | null |
HasCompactSupport.smul_right (hf : HasCompactSupport f) : HasCompactSupport (f • f') := by
rw [hasCompactSupport_iff_eventuallyEq] at hf ⊢
exact hf.mono fun x hx => by simp_rw [Pi.smul_apply', hx, Pi.zero_apply, zero_smul] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | HasCompactSupport.smul_right | null |
HasCompactSupport.mul_right (hf : HasCompactSupport f) : HasCompactSupport (f * f') := by
rw [hasCompactSupport_iff_eventuallyEq] at hf ⊢
exact hf.mono fun x hx => by simp_rw [Pi.mul_apply, hx, Pi.zero_apply, zero_mul] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | HasCompactSupport.mul_right | null |
HasCompactSupport.mul_left (hf : HasCompactSupport f') : HasCompactSupport (f * f') := by
rw [hasCompactSupport_iff_eventuallyEq] at hf ⊢
exact hf.mono fun x hx => by simp_rw [Pi.mul_apply, hx, Pi.zero_apply, mul_zero] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | HasCompactSupport.mul_left | null |
protected HasCompactSupport.abs {f : α → β} (hf : HasCompactSupport f) :
HasCompactSupport |f| :=
hf.comp_left (g := abs) abs_zero | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | HasCompactSupport.abs | null |
@[to_additive /-- If a family of functions `f` has locally-finite support, subordinate to a family
of open sets, then for any point we can find a neighbourhood on which only finitely-many members of
`f` are non-zero. -/]
LocallyFinite.exists_finset_nhds_mulSupport_subset {U : ι → Set X} [One R] {f : ι → X → R}
(hlf... | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | LocallyFinite.exists_finset_nhds_mulSupport_subset | If a family of functions `f` has locally-finite multiplicative support, subordinate to a family
of open sets, then for any point we can find a neighbourhood on which only finitely-many members of
`f` are not equal to 1. |
locallyFinite_mulSupport_iff [One M] {f : ι → X → M} :
(LocallyFinite fun i ↦ mulSupport <| f i) ↔ LocallyFinite fun i ↦ mulTSupport <| f i :=
⟨LocallyFinite.closure, fun H ↦ H.subset fun _ ↦ subset_closure⟩ | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | locallyFinite_mulSupport_iff | null |
LocallyFinite.smul_left [Zero R] [Zero M] [SMulWithZero R M]
{s : ι → X → R} (h : LocallyFinite fun i ↦ support <| s i) (f : ι → X → M) :
LocallyFinite fun i ↦ support <| s i • f i :=
h.subset fun i x ↦ mt <| fun h ↦ by rw [Pi.smul_apply', h, zero_smul] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | LocallyFinite.smul_left | null |
LocallyFinite.smul_right [Zero M] [SMulZeroClass R M]
{f : ι → X → M} (h : LocallyFinite fun i ↦ support <| f i) (s : ι → X → R) :
LocallyFinite fun i ↦ support <| s i • f i :=
h.subset fun i x ↦ mt <| fun h ↦ by rw [Pi.smul_apply', h, smul_zero] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | LocallyFinite.smul_right | null |
@[to_additive]
HasCompactMulSupport.comp_homeomorph {M} [One M] {f : Y → M}
(hf : HasCompactMulSupport f) (φ : X ≃ₜ Y) : HasCompactMulSupport (f ∘ φ) :=
hf.comp_isClosedEmbedding φ.isClosedEmbedding | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Order.Group.Unbundled.Abs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Algebra/Support.lean | HasCompactMulSupport.comp_homeomorph | null |
IsTopologicallyNilpotent
{R : Type*} [MonoidWithZero R] [TopologicalSpace R] (a : R) : Prop :=
Tendsto (a ^ ·) atTop (𝓝 0) | def | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | IsTopologicallyNilpotent | An element is topologically nilpotent if its powers converge to `0`. |
map {F : Type*} [FunLike F R S] [MonoidWithZeroHomClass F R S]
{φ : F} (hφ : Continuous φ) {a : R} (ha : IsTopologicallyNilpotent a) :
IsTopologicallyNilpotent (φ a) := by
unfold IsTopologicallyNilpotent at ha ⊢
simp_rw [← map_pow]
exact (map_zero φ ▸ hφ.tendsto 0).comp ha | theorem | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | map | The image of a topologically nilpotent element under a continuous morphism
is topologically nilpotent |
zero : IsTopologicallyNilpotent (0 : R) :=
tendsto_atTop_of_eventually_const (i₀ := 1)
(fun _ hi => by rw [zero_pow (Nat.ne_zero_iff_zero_lt.mpr hi)]) | theorem | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | zero | `0` is topologically nilpotent |
_root_.IsNilpotent.isTopologicallyNilpotent {a : R} (ha : IsNilpotent a) :
IsTopologicallyNilpotent a := by
obtain ⟨n, hn⟩ := ha
apply tendsto_atTop_of_eventually_const (i₀ := n)
intro i hi
rw [← Nat.add_sub_of_le hi, pow_add, hn, zero_mul] | theorem | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | _root_.IsNilpotent.isTopologicallyNilpotent | null |
exists_pow_mem_of_mem_nhds {a : R} (ha : IsTopologicallyNilpotent a)
{v : Set R} (hv : v ∈ 𝓝 0) :
∃ n, a ^ n ∈ v :=
(ha.eventually_mem hv).exists | theorem | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | exists_pow_mem_of_mem_nhds | null |
mul_right_of_commute [IsLinearTopology Rᵐᵒᵖ R]
{a b : R} (ha : IsTopologicallyNilpotent a) (hab : Commute a b) :
IsTopologicallyNilpotent (a * b) := by
simp_rw [IsTopologicallyNilpotent, hab.mul_pow]
exact IsLinearTopology.tendsto_mul_zero_of_left _ _ ha | theorem | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | mul_right_of_commute | If `a` and `b` commute and `a` is topologically nilpotent,
then `a * b` is topologically nilpotent. |
mul_left_of_commute [IsLinearTopology R R] {a b : R}
(hb : IsTopologicallyNilpotent b) (hab : Commute a b) :
IsTopologicallyNilpotent (a * b) := by
simp_rw [IsTopologicallyNilpotent, hab.mul_pow]
exact IsLinearTopology.tendsto_mul_zero_of_right _ _ hb | theorem | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | mul_left_of_commute | If `a` and `b` commute and `b` is topologically nilpotent,
then `a * b` is topologically nilpotent. |
add_of_commute [IsLinearTopology R R] {a b : R}
(ha : IsTopologicallyNilpotent a) (hb : IsTopologicallyNilpotent b) (h : Commute a b) :
IsTopologicallyNilpotent (a + b) := by
simp only [IsTopologicallyNilpotent, atTop_basis.tendsto_iff IsLinearTopology.hasBasis_ideal,
true_and]
intro I I_mem_nhds
obta... | theorem | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | add_of_commute | If `a` and `b` are topologically nilpotent and commute,
then `a + b` is topologically nilpotent. |
mul_right {a : R} (ha : IsTopologicallyNilpotent a) (b : R) :
IsTopologicallyNilpotent (a * b) :=
ha.mul_right_of_commute (Commute.all ..) | theorem | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | mul_right | If `a` is topologically nilpotent, then `a * b` is topologically nilpotent. |
mul_left (a : R) {b : R} (hb : IsTopologicallyNilpotent b) :
IsTopologicallyNilpotent (a * b) :=
hb.mul_left_of_commute (Commute.all ..) | theorem | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | mul_left | If `b` is topologically nilpotent, then `a * b` is topologically nilpotent. |
add {a b : R} (ha : IsTopologicallyNilpotent a) (hb : IsTopologicallyNilpotent b) :
IsTopologicallyNilpotent (a + b) :=
ha.add_of_commute hb (Commute.all ..)
variable (R) in | theorem | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | add | If `a` and `b` are topologically nilpotent, then `a + b` is topologically nilpotent. |
@[simps]
_root_.topologicalNilradical : Ideal R where
carrier := {a | IsTopologicallyNilpotent a}
add_mem' := add
zero_mem' := zero
smul_mem' := mul_left | def | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | _root_.topologicalNilradical | The topological nilradical of a ring with a linear topology |
mem_topologicalNilradical_iff {a : R} :
a ∈ topologicalNilradical R ↔ IsTopologicallyNilpotent a := by
simp [topologicalNilradical] | theorem | Topology | [
"Mathlib.Topology.Algebra.LinearTopology",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs"
] | Mathlib/Topology/Algebra/TopologicallyNilpotent.lean | mem_topologicalNilradical_iff | null |
@[to_additive (attr := simp)]
UniformFun.toFun_one [One β] : toFun (1 : α →ᵤ β) = 1 := rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.toFun_one | null |
UniformFun.ofFun_one [One β] : ofFun (1 : α → β) = 1 := rfl
@[to_additive] instance [One β] : One (α →ᵤ[𝔖] β) := Pi.instOne
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.ofFun_one | null |
UniformOnFun.toFun_one [One β] : toFun 𝔖 (1 : α →ᵤ[𝔖] β) = 1 := rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.toFun_one | null |
UniformOnFun.one_apply [One β] : ofFun 𝔖 (1 : α → β) = 1 := rfl
@[to_additive] instance [Mul β] : Mul (α →ᵤ β) := Pi.instMul
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.one_apply | null |
UniformFun.toFun_mul [Mul β] (f g : α →ᵤ β) : toFun (f * g) = toFun f * toFun g := rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.toFun_mul | null |
UniformFun.ofFun_mul [Mul β] (f g : α → β) : ofFun (f * g) = ofFun f * ofFun g := rfl
@[to_additive] instance [Mul β] : Mul (α →ᵤ[𝔖] β) := Pi.instMul
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.ofFun_mul | null |
UniformOnFun.toFun_mul [Mul β] (f g : α →ᵤ[𝔖] β) :
toFun 𝔖 (f * g) = toFun 𝔖 f * toFun 𝔖 g :=
rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.toFun_mul | null |
UniformOnFun.ofFun_mul [Mul β] (f g : α → β) : ofFun 𝔖 (f * g) = ofFun 𝔖 f * ofFun 𝔖 g := rfl
@[to_additive] instance [Inv β] : Inv (α →ᵤ β) := Pi.instInv
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.ofFun_mul | null |
UniformFun.toFun_inv [Inv β] (f : α →ᵤ β) : toFun (f⁻¹) = (toFun f)⁻¹ := rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.toFun_inv | null |
UniformFun.ofFun_inv [Inv β] (f : α → β) : ofFun (f⁻¹) = (ofFun f)⁻¹ := rfl
@[to_additive] instance [Inv β] : Inv (α →ᵤ[𝔖] β) := Pi.instInv
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.ofFun_inv | null |
UniformOnFun.toFun_inv [Inv β] (f : α →ᵤ[𝔖] β) : toFun 𝔖 (f⁻¹) = (toFun 𝔖 f)⁻¹ := rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.toFun_inv | null |
UniformOnFun.ofFun_inv [Inv β] (f : α → β) : ofFun 𝔖 (f⁻¹) = (ofFun 𝔖 f)⁻¹ := rfl
@[to_additive] instance [Div β] : Div (α →ᵤ β) := Pi.instDiv
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.ofFun_inv | null |
UniformFun.toFun_div [Div β] (f g : α →ᵤ β) : toFun (f / g) = toFun f / toFun g := rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.toFun_div | null |
UniformFun.ofFun_div [Div β] (f g : α → β) : ofFun (f / g) = ofFun f / ofFun g := rfl
@[to_additive] instance [Div β] : Div (α →ᵤ[𝔖] β) := Pi.instDiv
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.ofFun_div | null |
UniformOnFun.toFun_div [Div β] (f g : α →ᵤ[𝔖] β) :
toFun 𝔖 (f / g) = toFun 𝔖 f / toFun 𝔖 g :=
rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.toFun_div | null |
UniformOnFun.ofFun_div [Div β] (f g : α → β) : ofFun 𝔖 (f / g) = ofFun 𝔖 f / ofFun 𝔖 g := rfl
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.ofFun_div | null |
@[simp]
UniformFun.toFun_smul {M : Type*} [SMul M β] (c : M) (f : α →ᵤ β) :
toFun (c • f) = c • toFun f :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.toFun_smul | null |
UniformFun.ofFun_smul {M : Type*} [SMul M β] (c : M) (f : α → β) :
ofFun (c • f) = c • ofFun f :=
rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.ofFun_smul | null |
@[simp]
UniformOnFun.toFun_smul {M : Type*} [SMul M β] (c : M) (f : α →ᵤ[𝔖] β) :
toFun 𝔖 (c • f) = c • toFun 𝔖 f :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.toFun_smul | null |
UniformOnFun.ofFun_smul {M : Type*} [SMul M β] (c : M) (f : α → β) :
ofFun 𝔖 (c • f) = c • ofFun 𝔖 f :=
rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.ofFun_smul | null |
@[to_additive]
protected UniformFun.hasBasis_nhds_one_of_basis {p : ι → Prop} {b : ι → Set G}
(h : (𝓝 1 : Filter G).HasBasis p b) :
(𝓝 1 : Filter (α →ᵤ G)).HasBasis p fun i => { f : α →ᵤ G | ∀ x, toFun f x ∈ b i } := by
convert UniformFun.hasBasis_nhds_of_basis α _ (1 : α →ᵤ G) h.uniformity_of_nhds_one
si... | theorem | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.hasBasis_nhds_one_of_basis | null |
protected UniformFun.hasBasis_nhds_one :
(𝓝 1 : Filter (α →ᵤ G)).HasBasis (fun V : Set G => V ∈ (𝓝 1 : Filter G)) fun V =>
{ f : α → G | ∀ x, f x ∈ V } :=
UniformFun.hasBasis_nhds_one_of_basis (basis_sets _) | theorem | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformFun.hasBasis_nhds_one | null |
@[to_additive]
protected UniformOnFun.hasBasis_nhds_one_of_basis (𝔖 : Set <| Set α) (h𝔖₁ : 𝔖.Nonempty)
(h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) {p : ι → Prop} {b : ι → Set G}
(h : (𝓝 1 : Filter G).HasBasis p b) :
(𝓝 1 : Filter (α →ᵤ[𝔖] G)).HasBasis (fun Si : Set α × ι => Si.1 ∈ 𝔖 ∧ p Si.2) fun Si =>
{ f ... | theorem | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.hasBasis_nhds_one_of_basis | null |
protected UniformOnFun.hasBasis_nhds_one (𝔖 : Set <| Set α) (h𝔖₁ : 𝔖.Nonempty)
(h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) :
(𝓝 1 : Filter (α →ᵤ[𝔖] G)).HasBasis
(fun SV : Set α × Set G => SV.1 ∈ 𝔖 ∧ SV.2 ∈ (𝓝 1 : Filter G)) fun SV =>
{ f : α →ᵤ[𝔖] G | ∀ x ∈ SV.1, f x ∈ SV.2 } :=
UniformOnFun.hasBasis_n... | theorem | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.hasBasis_nhds_one | null |
UniformOnFun.ofFun_prod {β : Type*} [CommMonoid β] {f : ι → α → β} (I : Finset ι) :
ofFun 𝔖 (∏ i ∈ I, f i) = ∏ i ∈ I, ofFun 𝔖 (f i) :=
rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.ofFun_prod | null |
UniformOnFun.toFun_prod {β : Type*} [CommMonoid β] {f : ι → α → β} (I : Finset ι) :
toFun 𝔖 (∏ i ∈ I, f i) = ∏ i ∈ I, toFun 𝔖 (f i) :=
rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/Algebra/UniformConvergence.lean | UniformOnFun.toFun_prod | null |
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