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 ⌀ |
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instZeroHomClass : ZeroHomClass C(X, R)₀ X R where
map_zero f := f.map_zero' | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instZeroHomClass | null |
_root_.Set.zeroOfFactMem {X : Type*} [Zero X] (s : Set X) [Fact (0 ∈ s)] :
Zero s where
zero := ⟨0, Fact.out⟩
scoped[ContinuousMapZero] attribute [instance] Set.zeroOfFactMem
@[ext] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | _root_.Set.zeroOfFactMem | not marked as an instance because it would be a bad one in general, but it can
be useful when working with `ContinuousMapZero` and the non-unital continuous
functional calculus. |
ext {f g : C(X, R)₀} (h : ∀ x, f x = g x) : f = g := DFunLike.ext f g h
@[simp] | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | ext | null |
coe_mk {f : C(X, R)} {h0 : f 0 = 0} : ⇑(mk f h0) = f := rfl | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | coe_mk | null |
toContinuousMap_injective : Injective ((↑) : C(X, R)₀ → C(X, R)) :=
fun _ _ h ↦ congr(.mk $(h) _) | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | toContinuousMap_injective | null |
range_toContinuousMap : range ((↑) : C(X, R)₀ → C(X, R)) = {f : C(X, R) | f 0 = 0} :=
Set.ext fun f ↦ ⟨fun ⟨f', hf'⟩ ↦ hf' ▸ map_zero f', fun hf ↦ ⟨⟨f, hf⟩, rfl⟩⟩ | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | range_toContinuousMap | null |
comp (g : C(Y, R)₀) (f : C(X, Y)₀) : C(X, R)₀ where
toContinuousMap := (g : C(Y, R)).comp (f : C(X, Y))
map_zero' := show g (f 0) = 0 from map_zero f ▸ map_zero g
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | comp | Composition of continuous maps which map zero to zero. |
comp_apply (g : C(Y, R)₀) (f : C(X, Y)₀) (x : X) : g.comp f x = g (f x) := rfl | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | comp_apply | null |
instPartialOrder [PartialOrder R] : PartialOrder C(X, R)₀ :=
.lift _ DFunLike.coe_injective' | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instPartialOrder | null |
le_def [PartialOrder R] (f g : C(X, R)₀) : f ≤ g ↔ ∀ x, f x ≤ g x := Iff.rfl | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | le_def | null |
protected instTopologicalSpace : TopologicalSpace C(X, R)₀ :=
TopologicalSpace.induced ((↑) : C(X, R)₀ → C(X, R)) inferInstance | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instTopologicalSpace | null |
isEmbedding_toContinuousMap : IsEmbedding ((↑) : C(X, R)₀ → C(X, R)) where
eq_induced := rfl
injective _ _ h := ext fun x ↦ congr($(h) x) | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | isEmbedding_toContinuousMap | null |
instContinuousEvalConst : ContinuousEvalConst C(X, R)₀ X R :=
.of_continuous_forget isEmbedding_toContinuousMap.continuous | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instContinuousEvalConst | null |
instContinuousEval [LocallyCompactPair X R] : ContinuousEval C(X, R)₀ X R :=
.of_continuous_forget isEmbedding_toContinuousMap.continuous | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instContinuousEval | null |
isClosedEmbedding_toContinuousMap [T1Space R] :
IsClosedEmbedding ((↑) : C(X, R)₀ → C(X, R)) where
toIsEmbedding := isEmbedding_toContinuousMap
isClosed_range := by
rw [range_toContinuousMap]
exact isClosed_singleton.preimage <| continuous_eval_const 0
@[fun_prop] | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | isClosedEmbedding_toContinuousMap | null |
continuous_comp_left {X Y Z : Type*} [TopologicalSpace X]
[TopologicalSpace Y] [TopologicalSpace Z] [Zero X] [Zero Y] [Zero Z] (f : C(X, Y)₀) :
Continuous fun g : C(Y, Z)₀ ↦ g.comp f := by
rw [continuous_induced_rng]
change Continuous fun g : C(Y, Z)₀ ↦ (g : C(Y, Z)).comp (f : C(X, Y))
fun_prop | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | continuous_comp_left | null |
@[simps!]
protected id (s : Set R) [Fact (0 ∈ s)] : C(s, R)₀ :=
⟨.restrict s (.id R), rfl⟩
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | id | The identity function as an element of `C(s, R)₀` when `0 ∈ (s : Set R)`. |
toContinuousMap_id {s : Set R} [Fact (0 ∈ s)] :
(ContinuousMapZero.id s : C(s, R)) = .restrict s (.id R) :=
rfl | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | toContinuousMap_id | null |
noncomputable mkD [Zero X] (f : X → R) (default : C(X, R)₀) : C(X, R)₀ :=
if h : Continuous f ∧ f 0 = 0 then ⟨⟨_, h.1⟩, h.2⟩ else default | def | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | mkD | Interpret `f : α → β` as an element of `C(α, β)₀`, falling back to the default value
`default : C(α, β)₀` if `f` is not continuous or does not map `0` to `0`.
This is mainly intended to be used for `C(α, β)₀`-valued integration. For example, if a family of
functions `f : ι → α → β` satisfies that `f i` is continuous and maps `0` to `0` for almost every
`i`, you can write the `C(α, β)₀`-valued integral "`∫ i, f i`" as
`∫ i, ContinuousMapZero.mkD (f i) 0`. |
mkD_of_continuous [Zero X] {f : X → R} {g : C(X, R)₀} (hf : Continuous f) (hf₀ : f 0 = 0) :
mkD f g = ⟨⟨f, hf⟩, hf₀⟩ := by
simp only [mkD, And.intro hf hf₀, true_and, ↓reduceDIte] | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | mkD_of_continuous | null |
mkD_of_not_continuous [Zero X] {f : X → R} {g : C(X, R)₀} (hf : ¬ Continuous f) :
mkD f g = g := by
simp only [mkD, not_and_of_not_left _ hf, ↓reduceDIte] | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | mkD_of_not_continuous | null |
mkD_of_not_zero [Zero X] {f : X → R} {g : C(X, R)₀} (hf : f 0 ≠ 0) :
mkD f g = g := by
simp only [mkD, not_and_of_not_right _ hf, ↓reduceDIte] | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | mkD_of_not_zero | null |
mkD_apply_of_continuous [Zero X] {f : X → R} {g : C(X, R)₀} {x : X}
(hf : Continuous f) (hf₀ : f 0 = 0) :
mkD f g x = f x := by
rw [mkD_of_continuous hf hf₀, coe_mk, ContinuousMap.coe_mk] | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | mkD_apply_of_continuous | null |
mkD_of_continuousOn {s : Set X} [Zero s] {f : X → R} {g : C(s, R)₀}
(hf : ContinuousOn f s) (hf₀ : f (0 : s) = 0) :
mkD (s.restrict f) g = ⟨⟨s.restrict f, hf.restrict⟩, hf₀⟩ :=
mkD_of_continuous hf.restrict hf₀ | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | mkD_of_continuousOn | null |
mkD_of_not_continuousOn {s : Set X} [Zero s] {f : X → R} {g : C(s, R)₀}
(hf : ¬ ContinuousOn f s) :
mkD (s.restrict f) g = g := by
rw [continuousOn_iff_continuous_restrict] at hf
exact mkD_of_not_continuous hf | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | mkD_of_not_continuousOn | null |
mkD_apply_of_continuousOn {s : Set X} [Zero s] {f : X → R} {g : C(s, R)₀} {x : s}
(hf : ContinuousOn f s) (hf₀ : f (0 : s) = 0) :
mkD (s.restrict f) g x = f x := by
rw [mkD_of_continuousOn hf hf₀, coe_mk, ContinuousMap.coe_mk, restrict_apply]
open ContinuousMap in | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | mkD_apply_of_continuousOn | null |
mkD_eq_mkD_of_map_zero [Zero X] (f : X → R) (g : C(X, R)₀) (f_zero : f 0 = 0) :
mkD f g = ContinuousMap.mkD f g := by
ext
by_cases f_cont : Continuous f <;>
simp [*, ContinuousMap.mkD_of_continuous, mkD_of_continuous, mkD_of_not_continuous,
ContinuousMap.mkD_of_not_continuous] | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | mkD_eq_mkD_of_map_zero | Link between `ContinuousMapZero.mkD` and `ContinuousMap.mkD`. |
mkD_eq_self [Zero X] {f g : C(X, R)₀} : mkD f g = f :=
mkD_of_continuous f.continuous (map_zero f) | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | mkD_eq_self | null |
instZero [Zero R] : Zero C(X, R)₀ where
zero := ⟨0, rfl⟩
@[simp] lemma coe_zero [Zero R] : ⇑(0 : C(X, R)₀) = 0 := rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instZero | null |
instAdd [AddZeroClass R] [ContinuousAdd R] : Add C(X, R)₀ where
add f g := ⟨f + g, by simp⟩
@[simp] lemma coe_add [AddZeroClass R] [ContinuousAdd R] (f g : C(X, R)₀) : ⇑(f + g) = f + g := rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instAdd | null |
instNeg [NegZeroClass R] [ContinuousNeg R] : Neg C(X, R)₀ where
neg f := ⟨- f, by simp⟩
@[simp] lemma coe_neg [NegZeroClass R] [ContinuousNeg R] (f : C(X, R)₀) : ⇑(-f) = -f := rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instNeg | null |
instSub [SubNegZeroMonoid R] [ContinuousSub R] : Sub C(X, R)₀ where
sub f g := ⟨f - g, by simp⟩
@[simp] lemma coe_sub [SubNegZeroMonoid R] [ContinuousSub R] (f g : C(X, R)₀) :
⇑(f - g) = f - g := rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instSub | null |
instMul [MulZeroClass R] [ContinuousMul R] : Mul C(X, R)₀ where
mul f g := ⟨f * g, by simp⟩
@[simp] lemma coe_mul [MulZeroClass R] [ContinuousMul R] (f g : C(X, R)₀) : ⇑(f * g) = f * g := rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instMul | null |
instSMul {M : Type*} [Zero R] [SMulZeroClass M R] [ContinuousConstSMul M R] :
SMul M C(X, R)₀ where
smul m f := ⟨m • f, by simp⟩
@[simp] lemma coe_smul {M : Type*} [Zero R] [SMulZeroClass M R] [ContinuousConstSMul M R]
(m : M) (f : C(X, R)₀) : ⇑(m • f) = m • f := rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instSMul | null |
instAddCommMonoid : AddCommMonoid C(X, R)₀ :=
fast_instance% toContinuousMap_injective.addCommMonoid _ rfl (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instAddCommMonoid | null |
instModule {M : Type*} [Semiring M] [Module M R] [ContinuousConstSMul M R] :
Module M C(X, R)₀ :=
fast_instance% toContinuousMap_injective.module M
{ toFun := _, map_add' := fun _ _ ↦ rfl, map_zero' := rfl } (fun _ _ ↦ rfl) | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instModule | null |
instSMulCommClass {M N : Type*} [SMulZeroClass M R] [ContinuousConstSMul M R]
[SMulZeroClass N R] [ContinuousConstSMul N R] [SMulCommClass M N R] :
SMulCommClass M N C(X, R)₀ where
smul_comm _ _ _ := ext fun _ ↦ smul_comm .. | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instSMulCommClass | null |
instIsScalarTower {M N : Type*} [SMulZeroClass M R] [ContinuousConstSMul M R]
[SMulZeroClass N R] [ContinuousConstSMul N R] [SMul M N] [IsScalarTower M N R] :
IsScalarTower M N C(X, R)₀ where
smul_assoc _ _ _ := ext fun _ ↦ smul_assoc .. | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instIsScalarTower | null |
instAddCommGroup : AddCommGroup C(X, R)₀ :=
fast_instance% toContinuousMap_injective.addCommGroup _ rfl (fun _ _ ↦ rfl) (fun _ ↦ rfl)
(fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instAddCommGroup | null |
instNonUnitalCommSemiring : NonUnitalCommSemiring C(X, R)₀ :=
fast_instance% toContinuousMap_injective.nonUnitalCommSemiring
_ rfl (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instNonUnitalCommSemiring | null |
instSMulCommClass' {M : Type*} [SMulZeroClass M R] [SMulCommClass M R R]
[ContinuousConstSMul M R] : SMulCommClass M C(X, R)₀ C(X, R)₀ where
smul_comm m f g := ext fun x ↦ smul_comm m (f x) (g x) | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instSMulCommClass' | null |
instIsScalarTower' {M : Type*} [SMulZeroClass M R] [IsScalarTower M R R]
[ContinuousConstSMul M R] : IsScalarTower M C(X, R)₀ C(X, R)₀ where
smul_assoc m f g := ext fun x ↦ smul_assoc m (f x) (g x) | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instIsScalarTower' | null |
instStarRing [StarRing R] [ContinuousStar R] : StarRing C(X, R)₀ where
star f := ⟨star f, by simp⟩
star_involutive _ := ext fun _ ↦ star_star _
star_mul _ _ := ext fun _ ↦ star_mul ..
star_add _ _ := ext fun _ ↦ star_add .. | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instStarRing | null |
instStarModule [StarRing R] {M : Type*} [SMulZeroClass M R] [ContinuousConstSMul M R]
[Star M] [StarModule M R] [ContinuousStar R] : StarModule M C(X, R)₀ where
star_smul r f := ext fun x ↦ star_smul r (f x)
@[simp] lemma coe_star [StarRing R] [ContinuousStar R] (f : C(X, R)₀) : ⇑(star f) = star ⇑f := rfl | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instStarModule | null |
instCanLift : CanLift C(X, R) C(X, R)₀ (↑) (fun f ↦ f 0 = 0) where
prf f hf := ⟨⟨f, hf⟩, rfl⟩ | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instCanLift | null |
@[simps]
toContinuousMapHom [StarRing R] [ContinuousStar R] : C(X, R)₀ →⋆ₙₐ[R] C(X, R) where
toFun f := f
map_smul' _ _ := rfl
map_zero' := rfl
map_add' _ _ := rfl
map_mul' _ _ := rfl
map_star' _ := rfl | def | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | toContinuousMapHom | The coercion `C(X, R)₀ → C(X, R)` bundled as a non-unital star algebra homomorphism. |
coe_toContinuousMapHom [StarRing R] [ContinuousStar R] :
⇑(toContinuousMapHom (X := X) (R := R)) = (↑) :=
rfl | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | coe_toContinuousMapHom | null |
@[simps]
toContinuousMapCLM (M : Type*) [Semiring M] [Module M R] [ContinuousConstSMul M R] :
C(X, R)₀ →L[M] C(X, R) where
toFun f := f
map_add' _ _ := rfl
map_smul' _ _ := rfl | def | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | toContinuousMapCLM | The coercion `C(X, R)₀ → C(X, R)` bundled as a continuous linear map. |
evalCLM (𝕜 : Type*) [Semiring 𝕜] [Module 𝕜 R] [ContinuousConstSMul 𝕜 R] (x : X) :
C(X, R)₀ →L[𝕜] R :=
(ContinuousMap.evalCLM 𝕜 x).comp (toContinuousMapCLM 𝕜)
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | evalCLM | The evaluation at a point, as a continuous linear map from `C(X, R)₀` to `R`. |
evalCLM_apply {𝕜 : Type*} [Semiring 𝕜] [Module 𝕜 R] [ContinuousConstSMul 𝕜 R]
(x : X) (f : C(X, R)₀) : evalCLM 𝕜 x f = f x := rfl | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | evalCLM_apply | null |
coeFnAddMonoidHom : C(X, R)₀ →+ X → R where
toFun f := f
map_zero' := coe_zero
map_add' f g := by simp
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | coeFnAddMonoidHom | Coercion to a function as an `AddMonoidHom`. Similar to `ContinuousMap.coeFnAddMonoidHom`. |
coeFnAddMonoidHom_apply (f : C(X, R)₀) : coeFnAddMonoidHom f = f := rfl
@[simp] lemma coe_sum {ι : Type*} (s : Finset ι)
(f : ι → C(X, R)₀) : ⇑(s.sum f) = s.sum (fun i => ⇑(f i)) :=
map_sum coeFnAddMonoidHom f s | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | coeFnAddMonoidHom_apply | null |
instNonUnitalCommRing : NonUnitalCommRing C(X, R)₀ :=
fast_instance% toContinuousMap_injective.nonUnitalCommRing _ rfl
(fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instNonUnitalCommRing | null |
protected instUniformSpace : UniformSpace C(X, R)₀ := .comap toContinuousMap inferInstance | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | instUniformSpace | null |
isUniformEmbedding_toContinuousMap :
IsUniformEmbedding ((↑) : C(X, R)₀ → C(X, R)) where
comap_uniformity := rfl
injective _ _ h := ext fun x ↦ congr($(h) x) | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | isUniformEmbedding_toContinuousMap | null |
isUniformEmbedding_comp {Y : Type*} [UniformSpace Y] [Zero Y] (g : C(Y, R)₀)
(hg : IsUniformEmbedding g) : IsUniformEmbedding (g.comp · : C(X, Y)₀ → C(X, R)₀) :=
isUniformEmbedding_toContinuousMap.of_comp_iff.mp <|
ContinuousMap.isUniformEmbedding_comp g.toContinuousMap hg |>.comp
isUniformEmbedding_toContinuousMap | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | isUniformEmbedding_comp | null |
_root_.UniformEquiv.arrowCongrLeft₀ {Y : Type*} [TopologicalSpace Y] [Zero Y] (f : X ≃ₜ Y)
(hf : f 0 = 0) : C(X, R)₀ ≃ᵤ C(Y, R)₀ where
toFun g := g.comp ⟨f.symm, (f.toEquiv.apply_eq_iff_eq_symm_apply.eq ▸ hf).symm⟩
invFun g := g.comp ⟨f, hf⟩
left_inv g := ext fun _ ↦ congrArg g <| f.left_inv _
right_inv g := ext fun _ ↦ congrArg g <| f.right_inv _
uniformContinuous_toFun := isUniformEmbedding_toContinuousMap.uniformContinuous_iff.mpr <|
ContinuousMap.uniformContinuous_comp_left (f.symm : C(Y, X)) |>.comp
isUniformEmbedding_toContinuousMap.uniformContinuous
uniformContinuous_invFun := isUniformEmbedding_toContinuousMap.uniformContinuous_iff.mpr <|
ContinuousMap.uniformContinuous_comp_left (f : C(X, Y)) |>.comp
isUniformEmbedding_toContinuousMap.uniformContinuous | def | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | _root_.UniformEquiv.arrowCongrLeft₀ | The uniform equivalence `C(X, R)₀ ≃ᵤ C(Y, R)₀` induced by a homeomorphism of the domains
sending `0 : X` to `0 : Y`. |
@[simps]
nonUnitalStarAlgHom_precomp (f : C(X, Y)₀) : C(Y, R)₀ →⋆ₙₐ[R] C(X, R)₀ where
toFun g := g.comp f
map_zero' := rfl
map_add' _ _ := rfl
map_mul' _ _ := rfl
map_star' _ := rfl
map_smul' _ _ := rfl
variable (X) in | def | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | nonUnitalStarAlgHom_precomp | The functor `C(·, R)₀` from topological spaces with zero (and `ContinuousMapZero` maps) to
non-unital star algebras. |
@[simps apply]
nonUnitalStarAlgHom_postcomp (φ : R →⋆ₙₐ[M] S) (hφ : Continuous φ) :
C(X, R)₀ →⋆ₙₐ[M] C(X, S)₀ where
toFun := .comp ⟨⟨φ, hφ⟩, by simp⟩
map_zero' := ext <| by simp
map_add' _ _ := ext <| by simp
map_mul' _ _ := ext <| by simp
map_star' _ := ext <| by simp [map_star]
map_smul' r f := ext <| by simp | def | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | nonUnitalStarAlgHom_postcomp | The functor `C(X, ·)₀` from non-unital topological star algebras (with non-unital continuous
star homomorphisms) to non-unital star algebras. |
isometry_toContinuousMap [MetricSpace R] [Zero R] :
Isometry (toContinuousMap : C(α, R)₀ → C(α, R)) :=
fun _ _ ↦ rfl | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | isometry_toContinuousMap | null |
norm_def [NormedAddCommGroup R] (f : C(α, R)₀) : ‖f‖ = ‖(f : C(α, R))‖ :=
rfl | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.ContinuousMap.Compact"
] | Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean | norm_def | null |
ContinuousMap (X Y : Type*) [TopologicalSpace X] [TopologicalSpace Y] where
/-- The function `X → Y` -/
protected toFun : X → Y
/-- Proposition that `toFun` is continuous -/
protected continuous_toFun : Continuous toFun := by continuity | structure | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | ContinuousMap | The type of continuous maps from `X` to `Y`.
When possible, instead of parametrizing results over `(f : C(X, Y))`,
you should parametrize over `{F : Type*} [ContinuousMapClass F X Y] (f : F)`.
When you extend this structure, make sure to extend `ContinuousMapClass`. |
ContinuousMapClass (F : Type*) (X Y : outParam Type*)
[TopologicalSpace X] [TopologicalSpace Y] [FunLike F X Y] : Prop where
/-- Continuity -/
map_continuous (f : F) : Continuous f | class | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | ContinuousMapClass | `C(X, Y)` is the type of continuous maps from `X` to `Y`. -/
notation "C(" X ", " Y ")" => ContinuousMap X Y
section
/-- `ContinuousMapClass F X Y` states that `F` is a type of continuous maps.
You should extend this class when you extend `ContinuousMap`. |
@[coe] toContinuousMap (f : F) : C(X, Y) := ⟨f, map_continuous f⟩ | def | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | toContinuousMap | Coerce a bundled morphism with a `ContinuousMapClass` instance to a `ContinuousMap`. |
instFunLike : FunLike C(X, Y) X Y where
coe := ContinuousMap.toFun
coe_injective' f g h := by cases f; cases g; congr | instance | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | instFunLike | null |
instContinuousMapClass : ContinuousMapClass C(X, Y) X Y where
map_continuous := ContinuousMap.continuous_toFun
@[simp] | instance | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | instContinuousMapClass | null |
toFun_eq_coe {f : C(X, Y)} : f.toFun = (f : X → Y) :=
rfl | theorem | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | toFun_eq_coe | null |
Simps.apply (f : C(X, Y)) : X → Y := f
initialize_simps_projections ContinuousMap (toFun → apply)
@[simp] | def | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | Simps.apply | See note [custom simps projection]. |
protected coe_coe {F : Type*} [FunLike F X Y] [ContinuousMapClass F X Y] (f : F) :
⇑(f : C(X, Y)) = f :=
rfl | theorem | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | coe_coe | null |
protected coe_apply {F : Type*} [FunLike F X Y] [ContinuousMapClass F X Y] (f : F) (x : X) :
(f : C(X, Y)) x = f x :=
rfl | theorem | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | coe_apply | null |
protected coe_injective' {F : Type*} [FunLike F X Y] [ContinuousMapClass F X Y] :
Injective (toContinuousMap : F → C(X, Y)) :=
.of_comp (f := DFunLike.coe) DFunLike.coe_injective
@[ext] | theorem | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | coe_injective' | Coercion to a `ContinuousMap` is injective.
The unprimed version `ContinuousMap.coe_injective`
is used for the coercion from `C(X, Y)` to `X → Y`. |
ext {f g : C(X, Y)} (h : ∀ a, f a = g a) : f = g :=
DFunLike.ext _ _ h | theorem | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | ext | null |
protected copy (f : C(X, Y)) (f' : X → Y) (h : f' = f) : C(X, Y) where
toFun := f'
continuous_toFun := h.symm ▸ f.continuous_toFun
@[simp] | def | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | copy | Copy of a `ContinuousMap` with a new `toFun` equal to the old one. Useful to fix definitional
equalities. |
coe_copy (f : C(X, Y)) (f' : X → Y) (h : f' = f) : ⇑(f.copy f' h) = f' :=
rfl | theorem | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | coe_copy | null |
copy_eq (f : C(X, Y)) (f' : X → Y) (h : f' = f) : f.copy f' h = f :=
DFunLike.ext' h | theorem | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | copy_eq | null |
protected continuous (f : C(X, Y)) : Continuous f :=
f.continuous_toFun | theorem | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | continuous | Deprecated. Use `map_continuous` instead. |
protected congr_fun {f g : C(X, Y)} (H : f = g) (x : X) : f x = g x :=
H ▸ rfl | theorem | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | congr_fun | Deprecated. Use `DFunLike.congr_fun` instead. |
protected congr_arg (f : C(X, Y)) {x y : X} (h : x = y) : f x = f y :=
h ▸ rfl | theorem | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | congr_arg | Deprecated. Use `DFunLike.congr_arg` instead. |
coe_injective : Function.Injective (DFunLike.coe : C(X, Y) → (X → Y)) :=
DFunLike.coe_injective
@[simp] | theorem | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | coe_injective | null |
coe_mk (f : X → Y) (h : Continuous f) : ⇑(⟨f, h⟩ : C(X, Y)) = f :=
rfl | theorem | Topology | [
"Mathlib.Data.FunLike.Basic",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.Lift",
"Mathlib.Topology.Defs.Basic"
] | Mathlib/Topology/ContinuousMap/Defs.lean | coe_mk | null |
idealOfSet (s : Set X) : Ideal C(X, R) where
carrier := {f : C(X, R) | ∀ x ∈ sᶜ, f x = 0}
add_mem' {f g} hf hg x hx := by simp [hf x hx, hg x hx, add_zero]
zero_mem' _ _ := rfl
smul_mem' c _ hf x hx := mul_zero (c x) ▸ congr_arg (fun y => c x * y) (hf x hx) | def | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | idealOfSet | Given a topological ring `R` and `s : Set X`, construct the ideal in `C(X, R)` of functions
which vanish on the complement of `s`. |
idealOfSet_closed [T2Space R] (s : Set X) :
IsClosed (idealOfSet R s : Set C(X, R)) := by
simp only [idealOfSet, Submodule.coe_set_mk, Set.setOf_forall]
exact isClosed_iInter fun x => isClosed_iInter fun _ =>
isClosed_eq (continuous_eval_const x) continuous_const
variable {R} | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | idealOfSet_closed | null |
mem_idealOfSet {s : Set X} {f : C(X, R)} :
f ∈ idealOfSet R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0 := by
convert Iff.rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | mem_idealOfSet | null |
notMem_idealOfSet {s : Set X} {f : C(X, R)} : f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0 := by
simp_rw [mem_idealOfSet]; push_neg; rfl
@[deprecated (since := "2025-05-23")] alias not_mem_idealOfSet := notMem_idealOfSet | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | notMem_idealOfSet | null |
setOfIdeal (I : Ideal C(X, R)) : Set X :=
{x : X | ∀ f ∈ I, (f : C(X, R)) x = 0}ᶜ | def | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | setOfIdeal | Given an ideal `I` of `C(X, R)`, construct the set of points for which every function in the
ideal vanishes on the complement. |
notMem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
x ∉ setOfIdeal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0 := by
rw [← Set.mem_compl_iff, setOfIdeal, compl_compl, Set.mem_setOf]
@[deprecated (since := "2025-05-23")] alias not_mem_setOfIdeal := notMem_setOfIdeal | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | notMem_setOfIdeal | null |
mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
x ∈ setOfIdeal I ↔ ∃ f ∈ I, (f : C(X, R)) x ≠ 0 := by
simp_rw [setOfIdeal, Set.mem_compl_iff, Set.mem_setOf]; push_neg; rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | mem_setOfIdeal | null |
setOfIdeal_open [T2Space R] (I : Ideal C(X, R)) : IsOpen (setOfIdeal I) := by
simp only [setOfIdeal, Set.setOf_forall, isOpen_compl_iff]
exact
isClosed_iInter fun f =>
isClosed_iInter fun _ => isClosed_eq (map_continuous f) continuous_const | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | setOfIdeal_open | null |
@[simps]
opensOfIdeal [T2Space R] (I : Ideal C(X, R)) : Opens X :=
⟨setOfIdeal I, setOfIdeal_open I⟩
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | opensOfIdeal | The open set `ContinuousMap.setOfIdeal I` realized as a term of `opens X`. |
setOfTop_eq_univ [Nontrivial R] : setOfIdeal (⊤ : Ideal C(X, R)) = Set.univ :=
Set.univ_subset_iff.mp fun _ _ => mem_setOfIdeal.mpr ⟨1, Submodule.mem_top, one_ne_zero⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | setOfTop_eq_univ | null |
idealOfEmpty_eq_bot : idealOfSet R (∅ : Set X) = ⊥ :=
Ideal.ext fun f => by
simp only [mem_idealOfSet, Set.compl_empty, Set.mem_univ, forall_true_left, Ideal.mem_bot,
DFunLike.ext_iff, zero_apply]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | idealOfEmpty_eq_bot | null |
mem_idealOfSet_compl_singleton (x : X) (f : C(X, R)) :
f ∈ idealOfSet R ({x}ᶜ : Set X) ↔ f x = 0 := by
simp only [mem_idealOfSet, compl_compl, Set.mem_singleton_iff, forall_eq]
variable (X R) | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | mem_idealOfSet_compl_singleton | null |
ideal_gc : GaloisConnection (setOfIdeal : Ideal C(X, R) → Set X) (idealOfSet R) := by
refine fun I s => ⟨fun h f hf => ?_, fun h x hx => ?_⟩
· by_contra h'
rcases notMem_idealOfSet.mp h' with ⟨x, hx, hfx⟩
exact hfx (notMem_setOfIdeal.mp (mt (@h x) hx) hf)
· obtain ⟨f, hf, hfx⟩ := mem_setOfIdeal.mp hx
by_contra hx'
exact notMem_idealOfSet.mpr ⟨x, hx', hfx⟩ (h hf) | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | ideal_gc | null |
exists_mul_le_one_eqOn_ge (f : C(X, ℝ≥0)) {c : ℝ≥0} (hc : 0 < c) :
∃ g : C(X, ℝ≥0), (∀ x : X, (g * f) x ≤ 1) ∧ {x : X | c ≤ f x}.EqOn (g * f) 1 :=
⟨{ toFun := (f ⊔ const X c)⁻¹
continuous_toFun :=
((map_continuous f).sup <| map_continuous _).inv₀ fun _ => (hc.trans_le le_sup_right).ne' },
fun x =>
(inv_mul_le_iff₀ (hc.trans_le le_sup_right)).mpr ((mul_one (f x ⊔ c)).symm ▸ le_sup_left),
fun x hx => by
simpa only [coe_const, mul_apply, coe_mk, Pi.inv_apply, Pi.sup_apply,
Function.const_apply, sup_eq_left.mpr (Set.mem_setOf.mp hx), ne_eq, Pi.one_apply]
using inv_mul_cancel₀ (hc.trans_le hx).ne' ⟩
variable [CompactSpace X] [T2Space X]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | exists_mul_le_one_eqOn_ge | An auxiliary lemma used in the proof of `ContinuousMap.idealOfSet_ofIdeal_eq_closure` which may
be useful on its own. |
idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
idealOfSet 𝕜 (setOfIdeal I) = I.closure := by
/- Since `idealOfSet 𝕜 (setOfIdeal I)` is closed and contains `I`, it contains `I.closure`.
For the reverse inclusion, given `f ∈ idealOfSet 𝕜 (setOfIdeal I)` and `(ε : ℝ≥0) > 0` it
suffices to show that `f` is within `ε` of `I`. -/
refine le_antisymm ?_
((idealOfSet_closed 𝕜 <| setOfIdeal I).closure_subset_iff.mpr fun f hf x hx =>
notMem_setOfIdeal.mp hx hf)
refine (fun f hf => Metric.mem_closure_iff.mpr fun ε hε => ?_)
lift ε to ℝ≥0 using hε.lt.le
replace hε := show (0 : ℝ≥0) < ε from hε
simp_rw [dist_nndist]
norm_cast
set t := {x : X | ε / 2 ≤ ‖f x‖₊}
have ht : IsClosed t := isClosed_le continuous_const (map_continuous f).nnnorm
have htI : Disjoint t (setOfIdeal I)ᶜ := by
refine Set.subset_compl_iff_disjoint_left.mp fun x hx => ?_
simpa only [t, Set.mem_setOf, Set.mem_compl_iff, not_le] using
(nnnorm_eq_zero.mpr (mem_idealOfSet.mp hf hx)).trans_lt (half_pos hε)
/- It suffices to produce `g : C(X, ℝ≥0)` which takes values in `[0,1]` and is constantly `1` on
`t` such that when composed with the natural embedding of `ℝ≥0` into `𝕜` lies in the ideal `I`.
Indeed, then `‖f - f * ↑g‖ ≤ ‖f * (1 - ↑g)‖ ≤ ⨆ ‖f * (1 - ↑g) x‖`. When `x ∉ t`, `‖f x‖ < ε / 2`
and `‖(1 - ↑g) x‖ ≤ 1`, and when `x ∈ t`, `(1 - ↑g) x = 0`, and clearly `f * ↑g ∈ I`. -/
suffices
∃ g : C(X, ℝ≥0), (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g ∈ I ∧ (∀ x, g x ≤ 1) ∧ t.EqOn g 1 by
obtain ⟨g, hgI, hg, hgt⟩ := this
refine ⟨f * (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g, I.mul_mem_left f hgI, ?_⟩
rw [nndist_eq_nnnorm]
refine (nnnorm_lt_iff _ hε).2 fun x => ?_
simp only [coe_sub, coe_mul, Pi.sub_apply, Pi.mul_apply]
by_cases hx : x ∈ t
· simpa only [hgt hx, comp_apply, Pi.one_apply, ContinuousMap.coe_coe, algebraMapCLM_apply,
map_one, mul_one, sub_self, nnnorm_zero] using hε
· refine lt_of_le_of_lt ?_ (half_lt_self hε)
have :=
calc
‖((1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x : 𝕜)‖₊ =
‖1 - algebraMap ℝ≥0 𝕜 (g x)‖₊ := by
simp only [coe_sub, coe_one, coe_comp, ContinuousMap.coe_coe, Pi.sub_apply,
Pi.one_apply, Function.comp_apply, algebraMapCLM_apply]
_ = ‖algebraMap ℝ≥0 𝕜 (1 - g x)‖₊ := by
simp only [Algebra.algebraMap_eq_smul_one, NNReal.smul_def, NNReal.coe_sub (hg x),
NNReal.coe_one, sub_smul, one_smul]
_ ≤ 1 := (nnnorm_algebraMap_nnreal 𝕜 (1 - g x)).trans_le tsub_le_self
calc
‖f x - f x * (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g x‖₊ =
‖f x * (1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ := by
simp only [mul_sub, coe_sub, coe_one, Pi.sub_apply, Pi.one_apply, mul_one]
_ ≤ ε / 2 * ‖(1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
((nnnorm_mul_le _ _).trans
(mul_le_mul_right' (not_le.mp <| show ¬ε / 2 ≤ ‖f x‖₊ from hx).le _))
... | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | idealOfSet_ofIdeal_eq_closure | null |
idealOfSet_ofIdeal_isClosed {I : Ideal C(X, 𝕜)} (hI : IsClosed (I : Set C(X, 𝕜))) :
idealOfSet 𝕜 (setOfIdeal I) = I :=
(idealOfSet_ofIdeal_eq_closure I).trans (Ideal.ext <| Set.ext_iff.mp hI.closure_eq)
variable (𝕜)
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | idealOfSet_ofIdeal_isClosed | null |
setOfIdeal_ofSet_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜 s) = interior s := by
refine
Set.Subset.antisymm
((setOfIdeal_open (idealOfSet 𝕜 s)).subset_interior_iff.mpr fun x hx =>
let ⟨f, hf, hfx⟩ := mem_setOfIdeal.mp hx
Set.notMem_compl_iff.mp (mt (@hf x) hfx))
fun x hx => ?_
rw [← compl_compl (interior s), ← closure_compl] at hx
simp_rw [mem_setOfIdeal, mem_idealOfSet]
/- Apply Urysohn's lemma to get `g : C(X, ℝ)` which is zero on `sᶜ` and `g x ≠ 0`, then compose
with the natural embedding `ℝ ↪ 𝕜` to produce the desired `f`. -/
obtain ⟨g, hgs, hgx : Set.EqOn g 1 {x}, -⟩ :=
exists_continuous_zero_one_of_isClosed isClosed_closure isClosed_singleton
(Set.disjoint_singleton_right.mpr hx)
exact
⟨⟨fun x => g x, continuous_ofReal.comp (map_continuous g)⟩, by
simpa only [coe_mk, ofReal_eq_zero] using fun x hx => hgs (subset_closure hx), by
simpa only [coe_mk, hgx (Set.mem_singleton x), Pi.one_apply, RCLike.ofReal_one] using
one_ne_zero⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | setOfIdeal_ofSet_eq_interior | null |
setOfIdeal_ofSet_of_isOpen {s : Set X} (hs : IsOpen s) : setOfIdeal (idealOfSet 𝕜 s) = s :=
(setOfIdeal_ofSet_eq_interior 𝕜 s).trans hs.interior_eq
variable (X) in | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | setOfIdeal_ofSet_of_isOpen | null |
@[simps]
idealOpensGI :
GaloisInsertion (opensOfIdeal : Ideal C(X, 𝕜) → Opens X) fun s => idealOfSet 𝕜 s where
choice I _ := opensOfIdeal I.closure
gc I s := ideal_gc X 𝕜 I s
le_l_u s := (setOfIdeal_ofSet_of_isOpen 𝕜 s.isOpen).ge
choice_eq I hI :=
congr_arg _ <|
Ideal.ext
(Set.ext_iff.mp
(isClosed_of_closure_subset <|
(idealOfSet_ofIdeal_eq_closure I ▸ hI : I.closure ≤ I)).closure_eq) | def | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | idealOpensGI | The Galois insertion `ContinuousMap.opensOfIdeal : Ideal C(X, 𝕜) → Opens X` and
`fun s ↦ ContinuousMap.idealOfSet ↑s`. |
idealOfSet_isMaximal_iff (s : Opens X) :
(idealOfSet 𝕜 (s : Set X)).IsMaximal ↔ IsCoatom s := by
rw [Ideal.isMaximal_def]
refine (idealOpensGI X 𝕜).isCoatom_iff (fun I hI => ?_) s
rw [← Ideal.isMaximal_def] at hI
exact idealOfSet_ofIdeal_isClosed inferInstance | theorem | Topology | [
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Analysis.Normed.Ring.Units",
"Mathlib.Topology.Algebra.Module.CharacterSpace"
] | Mathlib/Topology/ContinuousMap/Ideals.lean | idealOfSet_isMaximal_iff | null |
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