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 ⌀ |
|---|---|---|---|---|---|---|
instNonUnitalNormedRing [NonUnitalNormedRing R] : NonUnitalNormedRing (α →ᵇ R) where
__ := instNonUnitalSeminormedRing
__ := instNormedAddCommGroup | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instNonUnitalNormedRing | null |
instNonUnitalNormedCommRing [NonUnitalNormedCommRing R] :
NonUnitalNormedCommRing (α →ᵇ R) where
mul_comm := mul_comm | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instNonUnitalNormedCommRing | null |
@[simp]
coe_npowRec (f : α →ᵇ R) : ∀ n, ⇑(npowRec n f) = (⇑f) ^ n
| 0 => by rw [npowRec, pow_zero, coe_one]
| n + 1 => by rw [npowRec, pow_succ, coe_mul, coe_npowRec f n] | theorem | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | coe_npowRec | null |
hasNatPow : Pow (α →ᵇ R) ℕ where
pow f n :=
{ toContinuousMap := f.toContinuousMap ^ n
map_bounded' := by simpa [coe_npowRec] using (npowRec n f).map_bounded' } | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | hasNatPow | null |
@[simp, norm_cast]
coe_natCast (n : ℕ) : ((n : α →ᵇ R) : α → R) = n := rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | coe_natCast | null |
coe_ofNat (n : ℕ) [n.AtLeastTwo] :
((ofNat(n) : α →ᵇ R) : α → R) = ofNat(n) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | coe_ofNat | null |
@[simp, norm_cast]
coe_intCast (n : ℤ) : ((n : α →ᵇ R) : α → R) = n := rfl | theorem | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | coe_intCast | null |
instRing : Ring (α →ᵇ R) := fast_instance%
DFunLike.coe_injective.ring _ coe_zero coe_one coe_add coe_mul coe_neg coe_sub
(fun _ _ => coe_nsmul _ _) (fun _ _ => coe_zsmul _ _) (fun _ _ => coe_pow _ _) coe_natCast
coe_intCast | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instRing | null |
instSeminormedRing : SeminormedRing (α →ᵇ R) where
__ := instRing
__ := instNonUnitalSeminormedRing | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instSeminormedRing | null |
@[simps!]
protected _root_.RingHom.compLeftContinuousBounded (α : Type*)
[TopologicalSpace α] [SeminormedRing β] [SeminormedRing γ]
(g : β →+* γ) {C : NNReal} (hg : LipschitzWith C g) : (α →ᵇ β) →+* (α →ᵇ γ) :=
{ g.toMonoidHom.compLeftContinuousBounded α hg,
g.toAddMonoidHom.compLeftContinuousBounded α hg... | def | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | _root_.RingHom.compLeftContinuousBounded | Composition on the left by a (lipschitz-continuous) homomorphism of topological semirings, as a
`RingHom`. Similar to `RingHom.compLeftContinuous`. |
instNormedRing [NormedRing R] : NormedRing (α →ᵇ R) where
__ := instRing
__ := instNonUnitalNormedRing | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instNormedRing | null |
instCommRing [SeminormedCommRing R] : CommRing (α →ᵇ R) where
mul_comm _ _ := ext fun _ ↦ mul_comm _ _ | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instCommRing | null |
instSeminormedCommRing [SeminormedCommRing R] : SeminormedCommRing (α →ᵇ R) where
__ := instCommRing
__ := instNonUnitalSeminormedRing | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instSeminormedCommRing | null |
instNormedCommRing [NormedCommRing R] : NormedCommRing (α →ᵇ R) where
__ := instSeminormedCommRing
__ := instNormedAddCommGroup | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instNormedCommRing | null |
C : 𝕜 →+* α →ᵇ γ where
toFun := fun c : 𝕜 => const α ((algebraMap 𝕜 γ) c)
map_one' := ext fun _ => (algebraMap 𝕜 γ).map_one
map_mul' _ _ := ext fun _ => (algebraMap 𝕜 γ).map_mul _ _
map_zero' := ext fun _ => (algebraMap 𝕜 γ).map_zero
map_add' _ _ := ext fun _ => (algebraMap 𝕜 γ).map_add _ _ | def | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | C | `BoundedContinuousFunction.const` as a `RingHom`. |
instAlgebra : Algebra 𝕜 (α →ᵇ γ) where
algebraMap := C
commutes' _ _ := ext fun _ ↦ Algebra.commutes' _ _
smul_def' _ _ := ext fun _ ↦ Algebra.smul_def' _ _
@[simp] | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instAlgebra | null |
algebraMap_apply (k : 𝕜) (a : α) : algebraMap 𝕜 (α →ᵇ γ) k a = k • (1 : γ) := by
simp only [Algebra.algebraMap_eq_smul_one, coe_smul, coe_one, Pi.one_apply] | theorem | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | algebraMap_apply | null |
instNormedAlgebra : NormedAlgebra 𝕜 (α →ᵇ γ) where
__ := instAlgebra
__ := instNormedSpace
variable (𝕜) | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instNormedAlgebra | null |
@[simps!]
protected AlgHom.compLeftContinuousBounded
[NormedRing β] [NormedAlgebra 𝕜 β] [NormedRing γ] [NormedAlgebra 𝕜 γ]
(g : β →ₐ[𝕜] γ) {C : NNReal} (hg : LipschitzWith C g) : (α →ᵇ β) →ₐ[𝕜] (α →ᵇ γ) :=
{ g.toRingHom.compLeftContinuousBounded α hg with
commutes' := fun _ => DFunLike.ext _ _ fun _ =... | def | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | AlgHom.compLeftContinuousBounded | Composition on the left by a (lipschitz-continuous) homomorphism of topological `R`-algebras,
as an `AlgHom`. Similar to `AlgHom.compLeftContinuous`. |
@[simps]
toContinuousMapₐ : (α →ᵇ γ) →ₐ[𝕜] C(α, γ) where
toFun := (↑)
map_one' := rfl
map_mul' _ _ := rfl
map_zero' := rfl
map_add' _ _ := rfl
commutes' _ := rfl
@[simp] | def | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | toContinuousMapₐ | The algebra-homomorphism forgetting that a bounded continuous function is bounded. |
coe_toContinuousMapₐ (f : α →ᵇ γ) : (f.toContinuousMapₐ 𝕜 : α → γ) = f := rfl
variable {𝕜}
/-! ### Structure as normed module over scalar functions
If `β` is a normed `𝕜`-space, then we show that the space of bounded continuous
functions from `α` to `β` is naturally a module over the algebra of bounded continuous
fu... | theorem | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | coe_toContinuousMapₐ | null |
instSMul' : SMul (α →ᵇ 𝕜) (α →ᵇ β) where
smul f g :=
ofNormedAddCommGroup (fun x => f x • g x) (f.continuous.smul g.continuous) (‖f‖ * ‖g‖) fun x =>
calc
‖f x • g x‖ ≤ ‖f x‖ * ‖g x‖ := norm_smul_le _ _
_ ≤ ‖f‖ * ‖g‖ :=
mul_le_mul (f.norm_coe_le_norm _) (g.norm_coe_le_norm _) (norm... | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instSMul' | null |
instModule' : Module (α →ᵇ 𝕜) (α →ᵇ β) :=
Module.ofMinimalAxioms
(fun c _ _ => ext fun a => smul_add (c a) _ _)
(fun _ _ _ => ext fun _ => add_smul _ _ _)
(fun _ _ _ => ext fun _ => mul_smul _ _ _)
(fun f => ext fun x => one_smul 𝕜 (f x))
/- TODO: When `NormedModule` has been added to `Analy... | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instModule' | null |
instPartialOrder : PartialOrder (α →ᵇ β) :=
PartialOrder.lift (fun f => f.toFun) (by simp [Injective]) | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instPartialOrder | null |
instSup : Max (α →ᵇ β) where
max f g :=
{ toFun := f ⊔ g
continuous_toFun := f.continuous.sup g.continuous
map_bounded' := by
obtain ⟨C₁, hf⟩ := f.bounded
obtain ⟨C₂, hg⟩ := g.bounded
refine ⟨C₁ + C₂, fun x y ↦ ?_⟩
simp_rw [dist_eq_norm_sub] at hf hg ⊢
exact (no... | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instSup | null |
instInf : Min (α →ᵇ β) where
min f g :=
{ toFun := f ⊓ g
continuous_toFun := f.continuous.inf g.continuous
map_bounded' := by
obtain ⟨C₁, hf⟩ := f.bounded
obtain ⟨C₂, hg⟩ := g.bounded
refine ⟨C₁ + C₂, fun x y ↦ ?_⟩
simp_rw [dist_eq_norm_sub] at hf hg ⊢
exact (no... | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instInf | null |
instSemilatticeSup : SemilatticeSup (α →ᵇ β) := fast_instance%
DFunLike.coe_injective.semilatticeSup _ coe_sup | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instSemilatticeSup | null |
instSemilatticeInf : SemilatticeInf (α →ᵇ β) := fast_instance%
DFunLike.coe_injective.semilatticeInf _ coe_inf | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instSemilatticeInf | null |
instLattice : Lattice (α →ᵇ β) := fast_instance%
DFunLike.coe_injective.lattice _ coe_sup coe_inf
@[simp, norm_cast] lemma coe_abs (f : α →ᵇ β) : ⇑|f| = |⇑f| := rfl
@[simp, norm_cast] lemma coe_posPart (f : α →ᵇ β) : ⇑f⁺ = (⇑f)⁺ := rfl
@[simp, norm_cast] lemma coe_negPart (f : α →ᵇ β) : ⇑f⁻ = (⇑f)⁻ := rfl | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instLattice | null |
instHasSolidNorm : HasSolidNorm (α →ᵇ β) :=
{ solid := by
intro f g h
have i1 : ∀ t, ‖f t‖ ≤ ‖g t‖ := fun t => HasSolidNorm.solid (h t)
rw [norm_le (norm_nonneg _)]
exact fun t => (i1 t).trans (norm_coe_le_norm g t) } | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instHasSolidNorm | null |
instIsOrderedAddMonoid : IsOrderedAddMonoid (α →ᵇ β) :=
{ add_le_add_left := by
intro f g h₁ h t
simp only [ContinuousMap.toFun_eq_coe, coe_toContinuousMap, coe_add, Pi.add_apply,
add_le_add_iff_left]
exact h₁ _ } | instance | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | instIsOrderedAddMonoid | null |
nnrealPart (f : α →ᵇ ℝ) : α →ᵇ ℝ≥0 :=
BoundedContinuousFunction.comp _ (show LipschitzWith 1 Real.toNNReal from lipschitzWith_posPart) f
@[simp] | def | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | nnrealPart | The nonnegative part of a bounded continuous `ℝ`-valued function as a bounded
continuous `ℝ≥0`-valued function. |
nnrealPart_coeFn_eq (f : α →ᵇ ℝ) : ⇑f.nnrealPart = Real.toNNReal ∘ ⇑f := rfl | theorem | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | nnrealPart_coeFn_eq | null |
nnnorm (f : α →ᵇ ℝ) : α →ᵇ ℝ≥0 :=
BoundedContinuousFunction.comp _
(show LipschitzWith 1 fun x : ℝ => ‖x‖₊ from lipschitzWith_one_norm) f
@[simp] | def | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | nnnorm | The absolute value of a bounded continuous `ℝ`-valued function as a bounded
continuous `ℝ≥0`-valued function. |
nnnorm_coeFn_eq (f : α →ᵇ ℝ) : ⇑f.nnnorm = NNNorm.nnnorm ∘ ⇑f := rfl | theorem | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | nnnorm_coeFn_eq | null |
self_eq_nnrealPart_sub_nnrealPart_neg (f : α →ᵇ ℝ) :
⇑f = (↑) ∘ f.nnrealPart - (↑) ∘ (-f).nnrealPart := by
funext x
dsimp
simp only [max_zero_sub_max_neg_zero_eq_self] | theorem | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | self_eq_nnrealPart_sub_nnrealPart_neg | Decompose a bounded continuous function to its positive and negative parts. |
abs_self_eq_nnrealPart_add_nnrealPart_neg (f : α →ᵇ ℝ) :
abs ∘ ⇑f = (↑) ∘ f.nnrealPart + (↑) ∘ (-f).nnrealPart := by
funext x
dsimp
simp only [max_zero_add_max_neg_zero_eq_abs_self] | theorem | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | abs_self_eq_nnrealPart_add_nnrealPart_neg | Express the absolute value of a bounded continuous function in terms of its
positive and negative parts. |
add_norm_nonneg (f : α →ᵇ ℝ) :
0 ≤ f + const _ ‖f‖ := by
intro x
simp only [ContinuousMap.toFun_eq_coe, coe_toContinuousMap, coe_zero, Pi.zero_apply, coe_add,
const_apply, Pi.add_apply]
linarith [(abs_le.mp (norm_coe_le_norm f x)).1] | lemma | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | add_norm_nonneg | null |
norm_sub_nonneg (f : α →ᵇ ℝ) :
0 ≤ const _ ‖f‖ - f := by
intro x
simp only [ContinuousMap.toFun_eq_coe, coe_toContinuousMap, coe_zero, Pi.zero_apply, coe_sub,
const_apply, Pi.sub_apply, sub_nonneg]
linarith [(abs_le.mp (norm_coe_le_norm f x)).2] | lemma | Topology | [
"Mathlib.Algebra.Module.MinimalAxioms",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Analysis.Normed.Operator.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/ContinuousMap/Bounded/Normed.lean | norm_sub_nonneg | null |
instStarAddMonoid : StarAddMonoid (α →ᵇ β) where
star f := f.comp star starNormedAddGroupHom.lipschitz
star_involutive f := ext fun x => star_star (f x)
star_add f g := ext fun x => star_add (f x) (g x) | instance | Topology | [
"Mathlib.Analysis.CStarAlgebra.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.ContinuousMap.Star"
] | Mathlib/Topology/ContinuousMap/Bounded/Star.lean | instStarAddMonoid | null |
@[simp]
coe_star (f : α →ᵇ β) : ⇑(star f) = star (⇑f) := rfl
@[simp] | theorem | Topology | [
"Mathlib.Analysis.CStarAlgebra.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.ContinuousMap.Star"
] | Mathlib/Topology/ContinuousMap/Bounded/Star.lean | coe_star | The right-hand side of this equality can be parsed `star ∘ ⇑f` because of the
instance `Pi.instStarForAll`. Upon inspecting the goal, one sees `⊢ ↑(star f) = star ↑f`. |
star_apply (f : α →ᵇ β) (x : α) : star f x = star (f x) := rfl | theorem | Topology | [
"Mathlib.Analysis.CStarAlgebra.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.ContinuousMap.Star"
] | Mathlib/Topology/ContinuousMap/Bounded/Star.lean | star_apply | null |
instNormedStarGroup : NormedStarGroup (α →ᵇ β) where
norm_star_le f := by simp only [norm_eq, star_apply, norm_star, le_of_eq] | instance | Topology | [
"Mathlib.Analysis.CStarAlgebra.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.ContinuousMap.Star"
] | Mathlib/Topology/ContinuousMap/Bounded/Star.lean | instNormedStarGroup | null |
instStarModule : StarModule 𝕜 (α →ᵇ β) where
star_smul k f := ext fun x => star_smul k (f x) | instance | Topology | [
"Mathlib.Analysis.CStarAlgebra.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.ContinuousMap.Star"
] | Mathlib/Topology/ContinuousMap/Bounded/Star.lean | instStarModule | null |
instStarRing [NormedStarGroup β] : StarRing (α →ᵇ β) where
__ := instStarAddMonoid
star_mul f g := ext fun x ↦ star_mul (f x) (g x)
variable [CStarRing β] | instance | Topology | [
"Mathlib.Analysis.CStarAlgebra.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.ContinuousMap.Star"
] | Mathlib/Topology/ContinuousMap/Bounded/Star.lean | instStarRing | null |
instCStarRing : CStarRing (α →ᵇ β) where
norm_mul_self_le f := by
rw [← sq, ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _), norm_le (Real.sqrt_nonneg _)]
intro x
rw [Real.le_sqrt (norm_nonneg _) (norm_nonneg _), sq, ← CStarRing.norm_star_mul_self]
exact norm_coe_le_norm (star f * f) x | instance | Topology | [
"Mathlib.Analysis.CStarAlgebra.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.ContinuousMap.Star"
] | Mathlib/Topology/ContinuousMap/Bounded/Star.lean | instCStarRing | null |
@[simps!]
toContinuousMapStarₐ : (α →ᵇ β) →⋆ₐ[𝕜] C(α, β) := { toContinuousMapₐ 𝕜 with
map_star' _ := rfl }
@[simp] | def | Topology | [
"Mathlib.Analysis.CStarAlgebra.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.ContinuousMap.Star"
] | Mathlib/Topology/ContinuousMap/Bounded/Star.lean | toContinuousMapStarₐ | The ⋆-algebra-homomorphism forgetting that a bounded continuous function is bounded. |
coe_toContinuousMapStarₐ (f : α →ᵇ β) : (f.toContinuousMapStarₐ 𝕜 : α → β) = f := rfl | theorem | Topology | [
"Mathlib.Analysis.CStarAlgebra.Basic",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.ContinuousMap.Star"
] | Mathlib/Topology/ContinuousMap/Bounded/Star.lean | coe_toContinuousMapStarₐ | null |
@[nolint unusedArguments]
basicCell (n : ℕ) (_ : Unit) : ∂𝔻 n ⟶ 𝔻 n := diskBoundaryInclusion n | abbrev | Topology | [
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Category.TopCat.Sphere",
"Mathlib.AlgebraicTopology.RelativeCellComplex.Basic"
] | Mathlib/Topology/CWComplex/Abstract/Basic.lean | basicCell | For each `n : ℕ`, this is the family of morphisms which sends the unique
element of `Unit` to `diskBoundaryInclusion n : ∂𝔻 n ⟶ 𝔻 n`. |
RelativeCWComplex {X Y : TopCat.{u}} (f : X ⟶ Y) := RelativeCellComplex.{u} basicCell f | abbrev | Topology | [
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Category.TopCat.Sphere",
"Mathlib.AlgebraicTopology.RelativeCellComplex.Basic"
] | Mathlib/Topology/CWComplex/Abstract/Basic.lean | RelativeCWComplex | A relative CW-complex is a morphism `f : X ⟶ Y` equipped with data expressing
that `Y` identifies to the colimit of a functor `F : ℕ ⥤ TopCat` with that
`F.obj 0 ≅ X` and for any `n : ℕ`, `F.obj (n + 1)` is obtained from `F.obj n`
by attaching `n`-disks. |
CWComplex (X : TopCat.{u}) := RelativeCWComplex (initial.to X) | abbrev | Topology | [
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Category.TopCat.Sphere",
"Mathlib.AlgebraicTopology.RelativeCellComplex.Basic"
] | Mathlib/Topology/CWComplex/Abstract/Basic.lean | CWComplex | A CW-complex is a topological space such that `⊥_ _ ⟶ X` is a relative CW-complex. |
RelCWComplex.{u} {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) where
/-- The indexing type of the cells of dimension `n`. -/
cell (n : ℕ) : Type u
/-- The characteristic map of the `n`-cell given by the index `i`.
This map is a bijection when restricting to `ball 0 1`, where we consider `... | class | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex. | A CW complex of a topological space `X` relative to another subspace `D` is the data of its
*`n`-cells* `cell n i` for each `n : ℕ` along with *attaching maps* that satisfy a number of
properties with the most important being closure-finiteness (`mapsTo`) and weak topology
(`closed'`). Note that this definition require... |
CWComplex.{u} {X : Type u} [TopologicalSpace X] (C : Set X) where
/-- The indexing type of the cells of dimension `n`. -/
protected cell (n : ℕ) : Type u
/-- The characteristic map of the `n`-cell given by the index `i`.
This map is a bijection when restricting to `ball 0 1`, where we consider `(Fin n → ℝ)`
e... | class | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex. | Characterizing when a subspace `C` of a topological space `X` is a CW complex. Note that this
requires `C` to be closed. If `C` is not closed choose `X` to be `C`. |
@[simps -isSimp]
RelCWComplex.toCWComplex {X : Type*} [TopologicalSpace X] (C : Set X) [RelCWComplex C ∅] :
CWComplex C where
cell := cell C
map := map
source_eq := source_eq
continuousOn := continuousOn
continuousOn_symm := continuousOn_symm
pairwiseDisjoint' := pairwiseDisjoint'
mapsTo' := by simpa ... | def | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.toCWComplex | A relative CW complex with an empty base is an absolute CW complex. |
RelCWComplex.toCWComplex_eq {X : Type*} [TopologicalSpace X] (C : Set X)
[h : RelCWComplex C ∅] : (toCWComplex C).instRelCWComplex = h :=
rfl
variable {X : Type*} [t : TopologicalSpace X] {C D : Set X} | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.toCWComplex_eq | null |
RelCWComplex.openCell [RelCWComplex C D] (n : ℕ) (i : cell C n) : Set X := map n i '' ball 0 1 | def | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.openCell | The open `n`-cell given by the index `i`. Use this instead of `map n i '' ball 0 1` whenever
possible. |
RelCWComplex.closedCell [RelCWComplex C D] (n : ℕ) (i : cell C n) : Set X :=
map n i '' closedBall 0 1 | def | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.closedCell | The closed `n`-cell given by the index `i`. Use this instead of `map n i '' closedBall 0 1`
whenever possible. |
RelCWComplex.cellFrontier [RelCWComplex C D] (n : ℕ) (i : cell C n) : Set X :=
map n i '' sphere 0 1 | def | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.cellFrontier | The boundary of the `n`-cell given by the index `i`. Use this instead of `map n i '' sphere 0 1`
whenever possible. |
CWComplex.mapsTo [CWComplex C] (n : ℕ) (i : cell C n) : ∃ I : Π m, Finset (cell C m),
MapsTo (map n i) (sphere 0 1) (⋃ (m < n) (j ∈ I m), map m j '' closedBall 0 1) := by
have := RelCWComplex.mapsTo n i
simp_rw [empty_union] at this
exact this | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.mapsTo | null |
RelCWComplex.pairwiseDisjoint [RelCWComplex C D] :
(univ : Set (Σ n, cell C n)).PairwiseDisjoint (fun ni ↦ openCell ni.1 ni.2) :=
RelCWComplex.pairwiseDisjoint' | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.pairwiseDisjoint | null |
RelCWComplex.disjointBase [RelCWComplex C D] (n : ℕ) (i : cell C n) :
Disjoint (openCell n i) D :=
RelCWComplex.disjointBase' n i | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.disjointBase | null |
RelCWComplex.disjoint_openCell_of_ne [RelCWComplex C D] {n m : ℕ} {i : cell C n}
{j : cell C m} (ne : (⟨n, i⟩ : Σ n, cell C n) ≠ ⟨m, j⟩) :
Disjoint (openCell n i) (openCell m j) :=
pairwiseDisjoint (mem_univ _) (mem_univ _) ne | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.disjoint_openCell_of_ne | null |
RelCWComplex.cellFrontier_subset_base_union_finite_closedCell [RelCWComplex C D]
(n : ℕ) (i : cell C n) : ∃ I : Π m, Finset (cell C m), cellFrontier n i ⊆
D ∪ ⋃ (m < n) (j ∈ I m), closedCell m j := by
rcases mapsTo n i with ⟨I, hI⟩
use I
rw [mapsTo_iff_image_subset] at hI
exact hI | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.cellFrontier_subset_base_union_finite_closedCell | null |
CWComplex.cellFrontier_subset_finite_closedCell [CWComplex C] (n : ℕ) (i : cell C n) :
∃ I : Π m, Finset (cell C m), cellFrontier n i ⊆ ⋃ (m < n) (j ∈ I m), closedCell m j := by
rcases RelCWComplex.mapsTo n i with ⟨I, hI⟩
use I
rw [mapsTo_iff_image_subset, empty_union] at hI
exact hI | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.cellFrontier_subset_finite_closedCell | null |
RelCWComplex.union [RelCWComplex C D] : D ∪ ⋃ (n : ℕ) (j : cell C n), closedCell n j = C :=
RelCWComplex.union' | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.union | null |
CWComplex.union [CWComplex C] : ⋃ (n : ℕ) (j : cell C n), closedCell n j = C := by
have := RelCWComplex.union' (C := C)
rw [empty_union] at this
exact this | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.union | null |
RelCWComplex.openCell_subset_closedCell [RelCWComplex C D] (n : ℕ) (i : cell C n) :
openCell n i ⊆ closedCell n i := image_mono Metric.ball_subset_closedBall | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.openCell_subset_closedCell | null |
RelCWComplex.cellFrontier_subset_closedCell [RelCWComplex C D] (n : ℕ) (i : cell C n) :
cellFrontier n i ⊆ closedCell n i := image_mono Metric.sphere_subset_closedBall | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.cellFrontier_subset_closedCell | null |
RelCWComplex.cellFrontier_union_openCell_eq_closedCell [RelCWComplex C D] (n : ℕ)
(i : cell C n) : cellFrontier n i ∪ openCell n i = closedCell n i := by
rw [cellFrontier, openCell, closedCell, ← image_union]
congrm map n i '' ?_
exact sphere_union_ball | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.cellFrontier_union_openCell_eq_closedCell | null |
RelCWComplex.map_zero_mem_openCell [RelCWComplex C D] (n : ℕ) (i : cell C n) :
map n i 0 ∈ openCell n i := by
apply mem_image_of_mem
simp only [mem_ball, dist_self, zero_lt_one] | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.map_zero_mem_openCell | null |
RelCWComplex.map_zero_mem_closedCell [RelCWComplex C D] (n : ℕ) (i : cell C n) :
map n i 0 ∈ closedCell n i :=
openCell_subset_closedCell _ _ (map_zero_mem_openCell _ _) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.map_zero_mem_closedCell | null |
private RelCWComplex.subset_of_eq_union_iUnion [RelCWComplex C D] (I J : Π n, Set (cell C n))
(hIJ : D ∪ ⋃ (n : ℕ) (j : I n), openCell (C := C) n j =
D ∪ ⋃ (n : ℕ) (j : J n), openCell (C := C) n j) (n : ℕ) :
I n ⊆ J n := by
intro i hi
by_contra hJ
have h : openCell n i ⊆ D ∪ ⋃ n, ⋃ (j : J n), openCe... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.subset_of_eq_union_iUnion | This is an auxiliary lemma used to prove `RelCWComplex.eq_of_eq_union_iUnion`. |
RelCWComplex.eq_of_eq_union_iUnion [RelCWComplex C D] (I J : Π n, Set (cell C n))
(hIJ : D ∪ ⋃ (n : ℕ) (j : I n), openCell (C := C) n j =
D ∪ ⋃ (n : ℕ) (j : J n), openCell (C := C) n j) :
I = J := by
ext n x
exact ⟨fun h ↦ subset_of_eq_union_iUnion I J hIJ n h,
fun h ↦ subset_of_eq_union_iUnion J ... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.eq_of_eq_union_iUnion | null |
CWComplex.eq_of_eq_union_iUnion [CWComplex C] (I J : Π n, Set (cell C n))
(hIJ : ⋃ (n : ℕ) (j : I n), openCell (C := C) n j =
⋃ (n : ℕ) (j : J n), openCell (C := C) n j) :
I = J := by
apply RelCWComplex.eq_of_eq_union_iUnion
simp_rw [empty_union, hIJ] | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.eq_of_eq_union_iUnion | null |
RelCWComplex.isCompact_closedCell [RelCWComplex C D] {n : ℕ} {i : cell C n} :
IsCompact (closedCell n i) :=
(isCompact_closedBall _ _).image_of_continuousOn (continuousOn n i) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.isCompact_closedCell | null |
RelCWComplex.isClosed_closedCell [RelCWComplex C D] [T2Space X] {n : ℕ} {i : cell C n} :
IsClosed (closedCell n i) := isCompact_closedCell.isClosed | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.isClosed_closedCell | null |
RelCWComplex.isCompact_cellFrontier [RelCWComplex C D] {n : ℕ} {i : cell C n} :
IsCompact (cellFrontier n i) :=
(isCompact_sphere _ _).image_of_continuousOn ((continuousOn n i).mono sphere_subset_closedBall) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.isCompact_cellFrontier | null |
RelCWComplex.isClosed_cellFrontier [RelCWComplex C D] [T2Space X] {n : ℕ} {i : cell C n} :
IsClosed (cellFrontier n i) :=
isCompact_cellFrontier.isClosed | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.isClosed_cellFrontier | null |
RelCWComplex.closure_openCell_eq_closedCell [RelCWComplex C D] [T2Space X] {n : ℕ}
{j : cell C n} : closure (openCell n j) = closedCell n j := by
apply subset_antisymm (isClosed_closedCell.closure_subset_iff.2 (openCell_subset_closedCell n j))
rw [closedCell, ← closure_ball 0 (by exact one_ne_zero)]
apply Con... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.closure_openCell_eq_closedCell | null |
RelCWComplex.closed (C : Set X) {D : Set X} [RelCWComplex C D] [T2Space X] (A : Set X)
(asubc : A ⊆ C) :
IsClosed A ↔ (∀ n (j : cell C n), IsClosed (A ∩ closedCell n j)) ∧ IsClosed (A ∩ D) := by
refine ⟨?_, closed' A asubc⟩
exact fun closedA ↦ ⟨fun _ _ ↦ closedA.inter isClosed_closedCell, closedA.inter (isC... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.closed | null |
CWComplex.closed (C : Set X) [CWComplex C] [T2Space X] (A : Set X) (asubc : A ⊆ C) :
IsClosed A ↔ ∀ n (j : cell C n), IsClosed (A ∩ closedCell n j) := by
have := RelCWComplex.closed C A asubc
simp_all | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.closed | null |
RelCWComplex.closedCell_subset_complex [RelCWComplex C D] (n : ℕ) (j : cell C n) :
closedCell n j ⊆ C := by
simp_rw [← union]
exact subset_union_of_subset_right (subset_iUnion₂ _ _) _ | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.closedCell_subset_complex | null |
RelCWComplex.openCell_subset_complex [RelCWComplex C D] (n : ℕ) (j : cell C n) :
openCell n j ⊆ C :=
(openCell_subset_closedCell _ _).trans (closedCell_subset_complex _ _) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.openCell_subset_complex | null |
RelCWComplex.cellFrontier_subset_complex [RelCWComplex C D] (n : ℕ) (j : cell C n) :
cellFrontier n j ⊆ C :=
(cellFrontier_subset_closedCell n j).trans (closedCell_subset_complex n j) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.cellFrontier_subset_complex | null |
RelCWComplex.closedCell_zero_eq_singleton [RelCWComplex C D] {j : cell C 0} :
closedCell 0 j = {map 0 j ![]} := by
simp [closedCell, Matrix.empty_eq] | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.closedCell_zero_eq_singleton | null |
RelCWComplex.openCell_zero_eq_singleton [RelCWComplex C D] {j : cell C 0} :
openCell 0 j = {map 0 j ![]} := by
simp [openCell, Matrix.empty_eq] | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.openCell_zero_eq_singleton | null |
RelCWComplex.cellFrontier_zero_eq_empty [RelCWComplex C D] {j : cell C 0} :
cellFrontier 0 j = ∅ := by
simp [cellFrontier, sphere_eq_empty_of_subsingleton] | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.cellFrontier_zero_eq_empty | null |
RelCWComplex.base_subset_complex [RelCWComplex C D] : D ⊆ C := by
simp_rw [← union]
exact subset_union_left | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.base_subset_complex | null |
RelCWComplex.isClosed [T2Space X] [RelCWComplex C D] : IsClosed C := by
rw [closed C C (by rfl)]
constructor
· intros
rw [inter_eq_right.2 (closedCell_subset_complex _ _)]
exact isClosed_closedCell
· rw [inter_eq_right.2 base_subset_complex]
exact isClosedBase C | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.isClosed | null |
private RelCWComplex.iUnion_openCell_eq_iUnion_closedCell [RelCWComplex C D] (n : ℕ∞) :
D ∪ ⋃ (m : ℕ) (_ : m < n) (j : cell C m), openCell m j =
D ∪ ⋃ (m : ℕ) (_ : m < n) (j : cell C m), closedCell m j := by
apply subset_antisymm
· apply union_subset
· exact subset_union_left
· apply iUnion₂_subse... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.iUnion_openCell_eq_iUnion_closedCell | A helper lemma that is essentially the same as `RelCWComplex.iUnion_openCell_eq_skeletonLT`.
Use that lemma instead. |
RelCWComplex.union_iUnion_openCell_eq_complex [RelCWComplex C D] :
D ∪ ⋃ (n : ℕ) (j : cell C n), openCell n j = C := by
suffices D ∪ ⋃ n, ⋃ (j : cell C n), openCell n j =
D ∪ ⋃ (m : ℕ) (_ : m < (⊤ : ℕ∞)) (j : cell C m), closedCell m j by
simpa [union] using this
simp_rw [← RelCWComplex.iUnion_openCell... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.union_iUnion_openCell_eq_complex | null |
CWComplex.iUnion_openCell_eq_complex [CWComplex C] :
⋃ (n : ℕ) (j : cell C n), openCell n j = C := by
simpa using RelCWComplex.union_iUnion_openCell_eq_complex (C := C) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.iUnion_openCell_eq_complex | null |
RelCWComplex.eq_of_not_disjoint_openCell [RelCWComplex C D] {n : ℕ} {j : cell C n} {m : ℕ}
{i : cell C m} (h : ¬ Disjoint (openCell n j) (openCell m i)) :
(⟨n, j⟩ : (Σ n, cell C n)) = ⟨m, i⟩ := by
contrapose! h
exact disjoint_openCell_of_ne h | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.eq_of_not_disjoint_openCell | The contrapositive of `disjoint_openCell_of_ne`. |
RelCWComplex.disjoint_base_iUnion_openCell [RelCWComplex C D] :
Disjoint D (⋃ (n : ℕ) (j : cell C n), openCell n j) := by
simp_rw [disjoint_iff_inter_eq_empty, inter_iUnion, iUnion_eq_empty]
intro n i
rw [inter_comm, (disjointBase n i).inter_eq] | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.disjoint_base_iUnion_openCell | null |
RelCWComplex.isClosed_inter_cellFrontier_succ_of_le_isClosed_inter_closedCell
[RelCWComplex C D] [T2Space X] {A : Set X} {n : ℕ} (hn : ∀ m ≤ n, ∀ (j : cell C m),
IsClosed (A ∩ closedCell m j)) (j : cell C (n + 1)) (hD : IsClosed (A ∩ D)) :
IsClosed (A ∩ cellFrontier (n + 1) j) := by
obtain ⟨I, hI⟩ := cell... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.isClosed_inter_cellFrontier_succ_of_le_isClosed_inter_closedCell | If for all `m ≤ n` and every `i : cell C m` the intersection `A ∩ closedCell m j` is closed
and `A ∩ D` is closed then `A ∩ cellFrontier (n + 1) j` is closed for every
`j : cell C (n + 1)`. |
CWComplex.isClosed_inter_cellFrontier_succ_of_le_isClosed_inter_closedCell
[CWComplex C] [T2Space X] {A : Set X} {n : ℕ} (hn : ∀ m ≤ n, ∀ (j : cell C m),
IsClosed (A ∩ closedCell m j)) (j : cell C (n + 1)) :
IsClosed (A ∩ cellFrontier (n + 1) j) :=
RelCWComplex.isClosed_inter_cellFrontier_succ_of_le_isClo... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.isClosed_inter_cellFrontier_succ_of_le_isClosed_inter_closedCell | null |
RelCWComplex.isClosed_of_isClosed_inter_openCell_or_isClosed_inter_closedCell
[RelCWComplex C D] [T2Space X] {A : Set X} (hAC : A ⊆ C) (hDA : IsClosed (A ∩ D))
(h : ∀ n (_ : 0 < n), ∀ (j : cell C n),
IsClosed (A ∩ openCell n j) ∨ IsClosed (A ∩ closedCell n j)) : IsClosed A := by
rw [closed C A hAC]
refi... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.isClosed_of_isClosed_inter_openCell_or_isClosed_inter_closedCell | If for every cell either `A ∩ openCell n j` or `A ∩ closedCell n j` is closed then
`A` is closed. |
RelCWComplex.isClosed_of_disjoint_openCell_or_isClosed_inter_closedCell
[RelCWComplex C D] [T2Space X] {A : Set X} (hAC : A ⊆ C) (hDA : IsClosed (A ∩ D))
(h : ∀ n (_ : 0 < n), ∀ (j : cell C n),
Disjoint A (openCell n j) ∨ IsClosed (A ∩ closedCell n j)) : IsClosed A := by
apply isClosed_of_isClosed_inter_o... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.isClosed_of_disjoint_openCell_or_isClosed_inter_closedCell | If for every cell either `A ∩ openCell n j` is empty or `A ∩ closedCell n j` is closed then
`A` is closed. |
CWComplex.isClosed_of_isClosed_inter_openCell_or_isClosed_inter_closedCell
[CWComplex C] [T2Space X] {A : Set X} (hAC : A ⊆ C) (h : ∀ n (_ : 0 < n), ∀ (j : cell C n),
IsClosed (A ∩ openCell n j) ∨ IsClosed (A ∩ closedCell n j)) : IsClosed A :=
RelCWComplex.isClosed_of_isClosed_inter_openCell_or_isClosed_inter... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.isClosed_of_isClosed_inter_openCell_or_isClosed_inter_closedCell | If for every cell either `A ∩ openCell n j` or `A ∩ closedCell n j` is closed then
`A` is closed. |
CWComplex.isClosed_of_disjoint_openCell_or_isClosed_inter_closedCell
[CWComplex C] [T2Space X] {A : Set X} (hAC : A ⊆ C) (h : ∀ n (_ : 0 < n), ∀ (j : cell C n),
Disjoint A (openCell n j) ∨ IsClosed (A ∩ closedCell n j)) : IsClosed A :=
RelCWComplex.isClosed_of_disjoint_openCell_or_isClosed_inter_closedCell hA... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.isClosed_of_disjoint_openCell_or_isClosed_inter_closedCell | If for every cell either `A ∩ openCell n j` is empty or `A ∩ closedCell n j` is closed then
`A` is closed. |
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