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Found 147 declarations mentioning ContinuousMap.comp.

ContinuousMap.comp 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : C(β, γ)) (g : C(α, β)) : C(α, γ)
ContinuousMap.comp_id 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) : f.comp (ContinuousMap.id α) = f
ContinuousMap.id_comp 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) : (ContinuousMap.id β).comp f = f
ContinuousMap.const_comp 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (c : γ) (f : C(α, β)) : (ContinuousMap.const β c).comp f = ContinuousMap.const α c
ContinuousMap.comp_const 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : C(β, γ)) (b : β) : f.comp (ContinuousMap.const α b) = ContinuousMap.const α (f b)
ContinuousMap.comp_assoc 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (f : C(γ, δ)) (g : C(β, γ)) (h : C(α, β)) : (f.comp g).comp h = f.comp (g.comp h)
ContinuousMap.cancel_left 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : C(β, γ)} {g₁ g₂ : C(α, β)} (hf : Function.Injective ⇑f) : f.comp g₁ = f.comp g₂ ↔ g₁ = g₂
ContinuousMap.cancel_right 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f₁ f₂ : C(β, γ)} {g : C(α, β)} (hg : Function.Surjective ⇑g) : f₁.comp g = f₂.comp g ↔ f₁ = f₂
ContinuousMap.comp_apply 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : C(β, γ)) (g : C(α, β)) (a : α) : (f.comp g) a = f (g a)
ContinuousMap.coe_comp 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : C(β, γ)) (g : C(α, β)) : ⇑(f.comp g) = ⇑f ∘ ⇑g
ContinuousMap.subtypeVal_comp_inclusion 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} [TopologicalSpace α] {s t : Set α} (h : s ⊆ t) : (ContinuousMap.subtypeVal t).comp (ContinuousMap.inclusion h) = ContinuousMap.subtypeVal s
Homeomorph.symm_comp_toContinuousMap 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) : (↑f.symm).comp ↑f = ContinuousMap.id α
Homeomorph.toContinuousMap_comp_symm 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) : (↑f).comp ↑f.symm = ContinuousMap.id β
Topology.IsQuotientMap.lift_comp 📋 Mathlib.Topology.ContinuousMap.Basic
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : Topology.IsQuotientMap ⇑f) (g : C(X, Z)) (h : Function.FactorsThrough ⇑g ⇑f) : (hf.lift g h).comp f = g
Homeomorph.coe_trans 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : α ≃ₜ β) (g : β ≃ₜ γ) : ↑(f.trans g) = (↑g).comp ↑f
ContinuousMap.inclusion_comp_inclusion 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} [TopologicalSpace α] {r s t : Set α} (hst : s ⊆ t) (hrs : r ⊆ s) : (ContinuousMap.inclusion hst).comp (ContinuousMap.inclusion hrs) = ContinuousMap.inclusion ⋯
ContinuousMap.piEquiv_symm_apply 📋 Mathlib.Topology.ContinuousMap.Basic
{I : Type u_5} (A : Type u_6) (X : I → Type u_7) [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : C(A, (i : I) → X i)) (i : I) : (ContinuousMap.piEquiv A X).symm f i = (ContinuousMap.eval i).comp f
Homeomorph.continuousMapCongr_apply 📋 Mathlib.Topology.ContinuousMap.Basic
{X₁ : Type u_5} {X₂ : Type u_6} {Y₁ : Type u_7} {Y₂ : Type u_8} [TopologicalSpace X₁] [TopologicalSpace X₂] [TopologicalSpace Y₁] [TopologicalSpace Y₂] (e : X₁ ≃ₜ X₂) (e' : Y₁ ≃ₜ Y₂) (f : C(X₁, Y₁)) : (e.continuousMapCongr e') f = { toFun := ⇑e', continuous_toFun := ⋯ }.comp (f.comp { toFun := ⇑e.symm, continuous_toFun := ⋯ })
Homeomorph.continuousMapCongr_symm_apply 📋 Mathlib.Topology.ContinuousMap.Basic
{X₁ : Type u_5} {X₂ : Type u_6} {Y₁ : Type u_7} {Y₂ : Type u_8} [TopologicalSpace X₁] [TopologicalSpace X₂] [TopologicalSpace Y₁] [TopologicalSpace Y₂] (e : X₁ ≃ₜ X₂) (e' : Y₁ ≃ₜ Y₂) (g : C(X₂, Y₂)) : (e.continuousMapCongr e').symm g = { toFun := ⇑e'.symm, continuous_toFun := ⋯ }.comp (g.comp { toFun := ⇑e, continuous_toFun := ⋯ })
ContinuousMap.sigmaEquiv_symm_apply 📋 Mathlib.Topology.ContinuousMap.Basic
{I : Type u_5} (A : Type u_6) (X : I → Type u_7) [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : C((i : I) × X i, A)) (i : I) : (ContinuousMap.sigmaEquiv A X).symm f i = f.comp (ContinuousMap.sigmaMk i)
Topology.IsQuotientMap.liftEquiv_symm_apply_coe 📋 Mathlib.Topology.ContinuousMap.Basic
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : Topology.IsQuotientMap ⇑f) (g : C(Y, Z)) : ↑(hf.liftEquiv.symm g) = g.comp f
ContinuousMap.addLeft_add 📋 Mathlib.Topology.Algebra.Monoid
{X : Type u_6} [AddSemigroup X] [TopologicalSpace X] [SeparatelyContinuousAdd X] (x y : X) : ContinuousMap.addLeft (x + y) = (ContinuousMap.addLeft x).comp (ContinuousMap.addLeft y)
ContinuousMap.addRight_add 📋 Mathlib.Topology.Algebra.Monoid
{X : Type u_6} [AddSemigroup X] [TopologicalSpace X] [SeparatelyContinuousAdd X] (x y : X) : ContinuousMap.addRight (x + y) = (ContinuousMap.addRight y).comp (ContinuousMap.addRight x)
ContinuousMap.mulLeft_mul 📋 Mathlib.Topology.Algebra.Monoid
{X : Type u_6} [Semigroup X] [TopologicalSpace X] [SeparatelyContinuousMul X] (x y : X) : ContinuousMap.mulLeft (x * y) = (ContinuousMap.mulLeft x).comp (ContinuousMap.mulLeft y)
ContinuousMap.mulRight_mul 📋 Mathlib.Topology.Algebra.Monoid
{X : Type u_6} [Semigroup X] [TopologicalSpace X] [SeparatelyContinuousMul X] (x y : X) : ContinuousMap.mulRight (x * y) = (ContinuousMap.mulRight y).comp (ContinuousMap.mulRight x)
TopologicalSpace.Opens.comap_comp 📋 Mathlib.Topology.Sets.Opens
{α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (g : C(β, γ)) (f : C(α, β)) : TopologicalSpace.Opens.comap (g.comp f) = (TopologicalSpace.Opens.comap f).comp (TopologicalSpace.Opens.comap g)
TopologicalSpace.Opens.comap_comap 📋 Mathlib.Topology.Sets.Opens
{α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (g : C(β, γ)) (f : C(α, β)) (U : TopologicalSpace.Opens γ) : (TopologicalSpace.Opens.comap f) ((TopologicalSpace.Opens.comap g) U) = (TopologicalSpace.Opens.comap (g.comp f)) U
SpectralMap.coe_comp_continuousMap' 📋 Mathlib.Topology.Spectral.Hom
{α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : SpectralMap β γ) (g : SpectralMap α β) : ↑(f.comp g) = (↑f).comp ↑g
TopCat.hom_comp 📋 Mathlib.Topology.Category.TopCat.Basic
{X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) : TopCat.Hom.hom (CategoryTheory.CategoryStruct.comp f g) = (TopCat.Hom.hom g).comp (TopCat.Hom.hom f)
TopCat.ofHom_comp 📋 Mathlib.Topology.Category.TopCat.Basic
{X Y Z : Type u} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : C(X, Y)) (g : C(Y, Z)) : TopCat.ofHom (g.comp f) = CategoryTheory.CategoryStruct.comp (TopCat.ofHom f) (TopCat.ofHom g)
LocallyConstant.comap_comap 📋 Mathlib.Topology.LocallyConstant.Basic
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {W : Type u_5} [TopologicalSpace W] (f : C(W, X)) (g : C(X, Y)) (x : LocallyConstant Y Z) : LocallyConstant.comap f (LocallyConstant.comap g x) = LocallyConstant.comap (g.comp f) x
LocallyConstant.comap_comp 📋 Mathlib.Topology.LocallyConstant.Basic
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {W : Type u_5} [TopologicalSpace W] (f : C(W, X)) (g : C(X, Y)) : LocallyConstant.comap (g.comp f) = LocallyConstant.comap f ∘ LocallyConstant.comap g
NNReal.ContinuousMap.canLift 📋 Mathlib.Topology.UniformSpace.Real
{X : Type u_1} [TopologicalSpace X] : CanLift C(X, ℝ) C(X, NNReal) ContinuousMap.coeNNRealReal.comp fun f => ∀ (x : X), 0 ≤ f x
ContinuousMap.continuous_postcomp 📋 Mathlib.Topology.CompactOpen
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (g : C(Y, Z)) : Continuous g.comp
ContinuousMap.continuous_precomp 📋 Mathlib.Topology.CompactOpen
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : C(X, Y)) : Continuous fun g => g.comp f
ContinuousMap.postcomp_injective 📋 Mathlib.Topology.CompactOpen
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (g : C(Y, Z)) (hg : Function.Injective ⇑g) : Function.Injective g.comp
ContinuousMap.isEmbedding_postcomp 📋 Mathlib.Topology.CompactOpen
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (g : C(Y, Z)) (hg : Topology.IsEmbedding ⇑g) : Topology.IsEmbedding g.comp
ContinuousMap.isInducing_postcomp 📋 Mathlib.Topology.CompactOpen
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (g : C(Y, Z)) (hg : Topology.IsInducing ⇑g) : Topology.IsInducing g.comp
Continuous.compCM 📋 Mathlib.Topology.CompactOpen
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z] {X' : Type u_6} [TopologicalSpace X'] {g : X' → C(Y, Z)} {f : X' → C(X, Y)} (hg : Continuous g) (hf : Continuous f) : Continuous fun x => (g x).comp (f x)
ContinuousAt.compCM 📋 Mathlib.Topology.CompactOpen
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z] {X' : Type u_6} [TopologicalSpace X'] {a : X'} {g : X' → C(Y, Z)} {f : X' → C(X, Y)} (hg : ContinuousAt g a) (hf : ContinuousAt f a) : ContinuousAt (fun x => (g x).comp (f x)) a
ContinuousOn.compCM 📋 Mathlib.Topology.CompactOpen
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z] {X' : Type u_6} [TopologicalSpace X'] {g : X' → C(Y, Z)} {f : X' → C(X, Y)} {s : Set X'} (hg : ContinuousOn g s) (hf : ContinuousOn f s) : ContinuousOn (fun x => (g x).comp (f x)) s
ContinuousWithinAt.compCM 📋 Mathlib.Topology.CompactOpen
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z] {X' : Type u_6} [TopologicalSpace X'] {a : X'} {g : X' → C(Y, Z)} {f : X' → C(X, Y)} {s : Set X'} (hg : ContinuousWithinAt g s a) (hf : ContinuousWithinAt f s a) : ContinuousWithinAt (fun x => (g x).comp (f x)) s a
ContinuousMap.continuous_comp' 📋 Mathlib.Topology.CompactOpen
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z] : Continuous fun x => x.2.comp x.1
ContinuousMap.compRightContinuousMap_apply 📋 Mathlib.Topology.CompactOpen
{X : Type u_2} {Y : Type u_3} (Z : Type u_4) [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : C(X, Y)) (g : C(Y, Z)) : (ContinuousMap.compRightContinuousMap Z f) g = g.comp f
Filter.Tendsto.compCM 📋 Mathlib.Topology.CompactOpen
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z] {α : Type u_6} {l : Filter α} {g : α → C(Y, Z)} {g₀ : C(Y, Z)} {f : α → C(X, Y)} {f₀ : C(X, Y)} (hg : Filter.Tendsto g l (nhds g₀)) (hf : Filter.Tendsto f l (nhds f₀)) : Filter.Tendsto (fun a => (g a).comp (f a)) l (nhds (g₀.comp f₀))
ContinuousMap.uniformContinuous_comp_left 📋 Mathlib.Topology.UniformSpace.CompactConvergence
{α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {γ : Type u_1} [TopologicalSpace γ] (g : C(α, γ)) : UniformContinuous fun f => f.comp g
ContinuousMap.isUniformEmbedding_comp 📋 Mathlib.Topology.UniformSpace.CompactConvergence
{α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {δ : Type u_2} [UniformSpace δ] (g : C(β, δ)) (hg : IsUniformEmbedding ⇑g) : IsUniformEmbedding g.comp
ContinuousMap.isUniformInducing_comp 📋 Mathlib.Topology.UniformSpace.CompactConvergence
{α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {δ : Type u_2} [UniformSpace δ] (g : C(β, δ)) (hg : IsUniformInducing ⇑g) : IsUniformInducing g.comp
ContinuousMap.uniformContinuous_comp 📋 Mathlib.Topology.UniformSpace.CompactConvergence
{α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {δ : Type u_2} [UniformSpace δ] (g : C(β, δ)) (hg : UniformContinuous ⇑g) : UniformContinuous g.comp
ContinuousMap.uniformSpace_eq_iInf_precomp_of_cover 📋 Mathlib.Topology.UniformSpace.CompactConvergence
{α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {ι : Type u₃} {δ : ι → Type u_1} [(i : ι) → TopologicalSpace (δ i)] (φ : (i : ι) → C(δ i, α)) (h_proper : ∀ (i : ι), IsProperMap ⇑(φ i)) (h_lf : LocallyFinite fun i => Set.range ⇑(φ i)) (h_cover : ⋃ i, Set.range ⇑(φ i) = Set.univ) : inferInstance = ⨅ i, UniformSpace.comap (fun x => x.comp (φ i)) inferInstance
ContinuousMap.uniformSpace_eq_inf_precomp_of_cover 📋 Mathlib.Topology.UniformSpace.CompactConvergence
{α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {δ₁ : Type u_1} {δ₂ : Type u_2} [TopologicalSpace δ₁] [TopologicalSpace δ₂] (φ₁ : C(δ₁, α)) (φ₂ : C(δ₂, α)) (h_proper₁ : IsProperMap ⇑φ₁) (h_proper₂ : IsProperMap ⇑φ₂) (h_cover : Set.range ⇑φ₁ ∪ Set.range ⇑φ₂ = Set.univ) : inferInstance = UniformSpace.comap (fun x => x.comp φ₁) inferInstance ⊓ UniformSpace.comap (fun x => x.comp φ₂) inferInstance
ContinuousMap.one_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [One γ] (g : C(α, β)) : ContinuousMap.comp 1 g = 1
ContinuousMap.zero_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Zero γ] (g : C(α, β)) : ContinuousMap.comp 0 g = 0
ContinuousMap.comp_one 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [One β] (g : C(β, γ)) : g.comp 1 = ContinuousMap.const α (g 1)
ContinuousMap.comp_zero 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Zero β] (g : C(β, γ)) : g.comp 0 = ContinuousMap.const α (g 0)
ContinuousMap.inv_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Inv γ] [ContinuousInv γ] (f : C(β, γ)) (g : C(α, β)) : f⁻¹.comp g = (f.comp g)⁻¹
ContinuousMap.neg_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Neg γ] [ContinuousNeg γ] (f : C(β, γ)) (g : C(α, β)) : (-f).comp g = -f.comp g
ContinuousMap.zpow_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Group γ] [IsTopologicalGroup γ] (f : C(β, γ)) (z : ℤ) (g : C(α, β)) : (f ^ z).comp g = f.comp g ^ z
ContinuousMap.zsmul_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [AddGroup γ] [IsTopologicalAddGroup γ] (f : C(β, γ)) (z : ℤ) (g : C(α, β)) : (z • f).comp g = z • f.comp g
ContinuousMap.smul_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {R : Type u_3} {M : Type u_5} [TopologicalSpace M] [SMul R M] [ContinuousConstSMul R M] (r : R) (f : C(β, M)) (g : C(α, β)) : (r • f).comp g = r • f.comp g
ContinuousMap.vadd_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {R : Type u_3} {M : Type u_5} [TopologicalSpace M] [VAdd R M] [ContinuousConstVAdd R M] (r : R) (f : C(β, M)) (g : C(α, β)) : (r +ᵥ f).comp g = r +ᵥ f.comp g
ContinuousMap.nsmul_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [AddMonoid γ] [ContinuousAdd γ] (f : C(β, γ)) (n : ℕ) (g : C(α, β)) : (n • f).comp g = n • f.comp g
ContinuousMap.pow_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Monoid γ] [ContinuousMul γ] (f : C(β, γ)) (n : ℕ) (g : C(α, β)) : (f ^ n).comp g = f.comp g ^ n
ContinuousMap.add_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Add γ] [ContinuousAdd γ] (f₁ f₂ : C(β, γ)) (g : C(α, β)) : (f₁ + f₂).comp g = f₁.comp g + f₂.comp g
ContinuousMap.div_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Div γ] [ContinuousDiv γ] (f g : C(β, γ)) (h : C(α, β)) : (f / g).comp h = f.comp h / g.comp h
ContinuousMap.mul_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Mul γ] [ContinuousMul γ] (f₁ f₂ : C(β, γ)) (g : C(α, β)) : (f₁ * f₂).comp g = f₁.comp g * f₂.comp g
ContinuousMap.sub_comp 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Sub γ] [ContinuousSub γ] (f g : C(β, γ)) (h : C(α, β)) : (f - g).comp h = f.comp h - g.comp h
ContinuousMap.compAddMonoidHom'_apply 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {γ : Type u_3} [TopologicalSpace γ] [AddZeroClass γ] [ContinuousAdd γ] (g : C(α, β)) (f : C(β, γ)) : g.compAddMonoidHom' f = f.comp g
ContinuousMap.compMonoidHom'_apply 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {γ : Type u_3} [TopologicalSpace γ] [MulOneClass γ] [ContinuousMul γ] (g : C(α, β)) (f : C(β, γ)) : g.compMonoidHom' f = f.comp g
ContinuousMap.compRightAlgHom_apply 📋 Mathlib.Topology.ContinuousMap.Algebra
(R : Type u_2) [CommSemiring R] (A : Type u_3) [TopologicalSpace A] [Semiring A] [Algebra R A] [IsTopologicalSemiring A] {α : Type u_5} {β : Type u_6} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (g : C(β, A)) : (ContinuousMap.compRightAlgHom R A f) g = g.comp f
AddMonoidHom.compLeftContinuous_apply 📋 Mathlib.Topology.ContinuousMap.Algebra
(α : Type u_1) {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {γ : Type u_3} [AddMonoid β] [ContinuousAdd β] [TopologicalSpace γ] [AddMonoid γ] [ContinuousAdd γ] (g : β →+ γ) (hg : Continuous ⇑g) (f : C(α, β)) : (AddMonoidHom.compLeftContinuous α g hg) f = { toFun := ⇑g, continuous_toFun := hg }.comp f
MonoidHom.compLeftContinuous_apply 📋 Mathlib.Topology.ContinuousMap.Algebra
(α : Type u_1) {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {γ : Type u_3} [Monoid β] [ContinuousMul β] [TopologicalSpace γ] [Monoid γ] [ContinuousMul γ] (g : β →* γ) (hg : Continuous ⇑g) (f : C(α, β)) : (MonoidHom.compLeftContinuous α g hg) f = { toFun := ⇑g, continuous_toFun := hg }.comp f
ContinuousMap.compCLM_apply 📋 Mathlib.Topology.ContinuousMap.Algebra
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (R : Type u_3) (M : Type u_5) [TopologicalSpace M] [Semiring R] [AddCommMonoid M] [ContinuousAdd M] [Module R M] [ContinuousConstSMul R M] (f : C(α, β)) (g : C(β, M)) : (ContinuousMap.compCLM R M f) g = g.comp f
ContinuousMap.compStarAlgHom'_apply 📋 Mathlib.Topology.ContinuousMap.Star
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (𝕜 : Type u_4) [CommSemiring 𝕜] (A : Type u_5) [TopologicalSpace A] [Semiring A] [IsTopologicalSemiring A] [Star A] [ContinuousStar A] [Algebra 𝕜 A] (f : C(X, Y)) (g : C(Y, A)) : (ContinuousMap.compStarAlgHom' 𝕜 A f) g = g.comp f
ContinuousMap.compStarAlgHom'_comp 📋 Mathlib.Topology.ContinuousMap.Star
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (𝕜 : Type u_4) [CommSemiring 𝕜] (A : Type u_5) [TopologicalSpace A] [Semiring A] [IsTopologicalSemiring A] [Star A] [ContinuousStar A] [Algebra 𝕜 A] (g : C(Y, Z)) (f : C(X, Y)) : ContinuousMap.compStarAlgHom' 𝕜 A (g.comp f) = (ContinuousMap.compStarAlgHom' 𝕜 A f).comp (ContinuousMap.compStarAlgHom' 𝕜 A g)
ContinuousMap.compStarAlgHom_apply 📋 Mathlib.Topology.ContinuousMap.Star
(X : Type u_1) {𝕜 : Type u_2} {A : Type u_3} {B : Type u_4} [TopologicalSpace X] [CommSemiring 𝕜] [TopologicalSpace A] [Semiring A] [IsTopologicalSemiring A] [Star A] [ContinuousStar A] [Algebra 𝕜 A] [TopologicalSpace B] [Semiring B] [IsTopologicalSemiring B] [Star B] [ContinuousStar B] [Algebra 𝕜 B] (φ : A →⋆ₐ[𝕜] B) (hφ : Continuous ⇑φ) (f : C(X, A)) : (ContinuousMap.compStarAlgHom X φ hφ) f = { toFun := ⇑φ, continuous_toFun := hφ }.comp f
AlgebraicGeometry.continuousMapPresheaf_map 📋 Mathlib.AlgebraicGeometry.Sites.ConstantSheaf
(T : Type v) [TopologicalSpace T] {U V : AlgebraicGeometry.Schemeᵒᵖ} (f : U ⟶ V) : (AlgebraicGeometry.continuousMapPresheaf T).map f = TypeCat.ofHom fun g => g.comp (TopCat.Hom.hom f.unop.base)
ContinuousMap.Homotopy.compContinuousMap 📋 Mathlib.Topology.Homotopy.Basic
{X : Type u} {Y : Type v} {Z : Type w} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {g₀ g₁ : C(Y, Z)} (G : g₀.Homotopy g₁) (f : C(X, Y)) : (g₀.comp f).Homotopy (g₁.comp f)
ContinuousMap.HomotopicRel.comp_continuousMap 📋 Mathlib.Topology.Homotopy.Basic
{X : Type u} {Y : Type v} {Z : Type w} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {S : Set X} ⦃f₀ f₁ : C(X, Y)⦄ (h : f₀.HomotopicRel f₁ S) (g : C(Y, Z)) : (g.comp f₀).HomotopicRel (g.comp f₁) S
ContinuousMap.HomotopyRel.compContinuousMap 📋 Mathlib.Topology.Homotopy.Basic
{X : Type u} {Y : Type v} {Z : Type w} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {S : Set X} {f₀ f₁ : C(X, Y)} (F : f₀.HomotopyRel f₁ S) (g : C(Y, Z)) : (g.comp f₀).HomotopyRel (g.comp f₁) S
ContinuousMap.Homotopic.comp 📋 Mathlib.Topology.Homotopy.Basic
{X : Type u} {Y : Type v} {Z : Type w} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {g₀ g₁ : C(Y, Z)} {f₀ f₁ : C(X, Y)} (hg : g₀.Homotopic g₁) (hf : f₀.Homotopic f₁) : (g₀.comp f₀).Homotopic (g₁.comp f₁)
ContinuousMap.Homotopy.comp 📋 Mathlib.Topology.Homotopy.Basic
{X : Type u} {Y : Type v} {Z : Type w} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(Y, Z)} (G : g₀.Homotopy g₁) (F : f₀.Homotopy f₁) : (g₀.comp f₀).Homotopy (g₁.comp f₁)
ContinuousMap.Homotopy.compContinuousMap_apply 📋 Mathlib.Topology.Homotopy.Basic
{X : Type u} {Y : Type v} {Z : Type w} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {g₀ g₁ : C(Y, Z)} (G : g₀.Homotopy g₁) (f : C(X, Y)) (x : ↑unitInterval × X) : (G.compContinuousMap f) x = G (x.1, f x.2)
ContinuousMap.Homotopy.comp_apply 📋 Mathlib.Topology.Homotopy.Basic
{X : Type u} {Y : Type v} {Z : Type w} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(Y, Z)} (G : g₀.Homotopy g₁) (F : f₀.Homotopy f₁) (x : ↑unitInterval × X) : (G.comp F) x = G (x.1, F x)
ContinuousMap.HomotopyRel.compContinuousMap_apply 📋 Mathlib.Topology.Homotopy.Basic
{X : Type u} {Y : Type v} {Z : Type w} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {S : Set X} {f₀ f₁ : C(X, Y)} (F : f₀.HomotopyRel f₁ S) (g : C(Y, Z)) (x : ↑unitInterval × X) : (F.compContinuousMap g) x = g (F x)
Path.Homotopic.Quotient.map_comp 📋 Mathlib.Topology.Homotopy.Path
{X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} {Z : Type u_1} [TopologicalSpace Z] {p : Path.Homotopic.Quotient x₀ x₁} {f : C(X, Y)} {g : C(Y, Z)} : p.map (g.comp f) = (p.map f).map g
FundamentalGroupoid.map_comp 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_3} [TopologicalSpace Z] (g : C(Y, Z)) (f : C(X, Y)) : FundamentalGroupoid.map (g.comp f) = (FundamentalGroupoid.map f).comp (FundamentalGroupoid.map g)
ContinuousMap.HomotopyEquiv.left_inv 📋 Mathlib.Topology.Homotopy.Equiv
{X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (self : ContinuousMap.HomotopyEquiv X Y) : (self.invFun.comp self.toFun).Homotopic (ContinuousMap.id X)
ContinuousMap.HomotopyEquiv.right_inv 📋 Mathlib.Topology.Homotopy.Equiv
{X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (self : ContinuousMap.HomotopyEquiv X Y) : (self.toFun.comp self.invFun).Homotopic (ContinuousMap.id Y)
ContinuousMap.HomotopyEquiv.mk 📋 Mathlib.Topology.Homotopy.Equiv
{X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (toFun : C(X, Y)) (invFun : C(Y, X)) (left_inv : (invFun.comp toFun).Homotopic (ContinuousMap.id X)) (right_inv : (toFun.comp invFun).Homotopic (ContinuousMap.id Y)) : ContinuousMap.HomotopyEquiv X Y
ContinuousMap.Nullhomotopic.comp_left 📋 Mathlib.Topology.Homotopy.Contractible
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(Y, Z)} (hf : f.Nullhomotopic) (g : C(X, Y)) : (f.comp g).Nullhomotopic
ContinuousMap.Nullhomotopic.comp_right 📋 Mathlib.Topology.Homotopy.Contractible
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : f.Nullhomotopic) (g : C(Y, Z)) : (g.comp f).Nullhomotopic
TopCat.toSSetObjEquiv_symm_naturality 📋 Mathlib.AlgebraicTopology.SimplicialSet.TopAdj
{X : TopCat} {n m : SimplexCategory} (f : n ⟶ m) (g : C(Convexity.StdSimplex ℝ (Fin (m.len + 1)), ↑X)) : (CategoryTheory.ConcreteCategory.hom ((TopCat.toSSet.obj X).map f.op)) ((X.toSSetObjEquiv (Opposite.op m)).symm g) = (X.toSSetObjEquiv (Opposite.op n)).symm (g.comp { toFun := Convexity.StdSimplex.map ⇑(CategoryTheory.ConcreteCategory.hom f), continuous_toFun := ⋯ })
cfcHomSuperset_apply 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital
{R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] [instCFC : ContinuousFunctionalCalculus R A p] {a : A} (ha : p a) {s : Set R} (hs : spectrum R a ⊆ s) (x : C(↑s, R)) : (cfcHomSuperset ha hs) x = (cfcHom ha) (x.comp { toFun := Subtype.map id hs, continuous_toFun := ⋯ })
cfcHom_comp 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital
{R : Type u_1} {A : Type u_2} {p : A → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra R A] [instCFC : ContinuousFunctionalCalculus R A p] {a : A} (ha : p a) [ContinuousMap.UniqueHom R A] (f : C(↑(spectrum R a), R)) (f' : C(↑(spectrum R a), ↑(spectrum R ((cfcHom ha) f)))) (hff' : ∀ (x : ↑(spectrum R a)), f x = ↑(f' x)) (g : C(↑(spectrum R ((cfcHom ha) f)), R)) : (cfcHom ha) (g.comp f') = (cfcHom ⋯) g
SpectrumRestricts.starAlgHom_apply 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
{R : Type u} {S : Type v} {A : Type w} [Semifield R] [StarRing R] [TopologicalSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [Semifield S] [StarRing S] [TopologicalSpace S] [IsTopologicalSemiring S] [ContinuousStar S] [Ring A] [StarRing A] [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] {a : A} (φ : C(↑(spectrum S a), S) →⋆ₐ[S] A) {f : C(S, R)} (h : SpectrumRestricts a ⇑f) (x : C(↑(spectrum R a), R)) : (SpectrumRestricts.starAlgHom φ h) x = φ ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp (x.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
ContinuousMap.polynomial_comp_attachBound_mem 📋 Mathlib.Topology.ContinuousMap.StoneWeierstrass
{X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (f : ↥A) (g : Polynomial ℝ) : (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (↑f).attachBound ∈ A
ContinuousMap.comp_attachBound_mem_closure 📋 Mathlib.Topology.ContinuousMap.StoneWeierstrass
{X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (f : ↥A) (p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ)) : p.comp (↑f).attachBound ∈ A.topologicalClosure
ContinuousMap.polynomial_comp_attachBound 📋 Mathlib.Topology.ContinuousMap.StoneWeierstrass
{X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (f : ↥A) (g : Polynomial ℝ) : (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (↑f).attachBound = ↑((Polynomial.aeval f) g)
StarAlgHom.realContinuousMapOfNNReal_apply_comp_toReal 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique
{X : Type u_1} [TopologicalSpace X] {A : Type u_2} [Ring A] [StarRing A] [Algebra ℝ A] (φ : C(X, NNReal) →⋆ₐ[NNReal] A) (f : C(X, NNReal)) : φ.realContinuousMapOfNNReal ({ toFun := NNReal.toReal, continuous_toFun := NNReal.continuous_coe }.comp f) = φ f
WeakDual.CharacterSpace.compContinuousMap_comp 📋 Mathlib.Analysis.CStarAlgebra.GelfandDuality
{A : Type u_1} {B : Type u_2} {C : Type u_3} {𝕜 : Type u_4} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [StarRing A] [NormedRing B] [NormedAlgebra 𝕜 B] [CompleteSpace B] [StarRing B] [NormedRing C] [NormedAlgebra 𝕜 C] [CompleteSpace C] [StarRing C] (ψ₂ : B →⋆ₐ[𝕜] C) (ψ₁ : A →⋆ₐ[𝕜] B) : WeakDual.CharacterSpace.compContinuousMap (ψ₂.comp ψ₁) = (WeakDual.CharacterSpace.compContinuousMap ψ₁).comp (WeakDual.CharacterSpace.compContinuousMap ψ₂)
WeakDual.CharacterSpace.homeoEval_naturality 📋 Mathlib.Analysis.CStarAlgebra.GelfandDuality
{X : Type u_1} {Y : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] [TopologicalSpace X] [CompactSpace X] [T2Space X] [TopologicalSpace Y] [CompactSpace Y] [T2Space Y] (f : C(X, Y)) : (↑(WeakDual.CharacterSpace.homeoEval Y 𝕜)).comp f = (WeakDual.CharacterSpace.compContinuousMap (ContinuousMap.compStarAlgHom' 𝕜 𝕜 f)).comp ↑(WeakDual.CharacterSpace.homeoEval X 𝕜)
cfcHomTransfer_apply 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Transfer
{R : Type u_1} {A : Type u_2} {B : Type u_3} {p : A → Prop} {q : B → Prop} [CommSemiring R] [StarRing R] [MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A] [Ring B] [StarRing B] [Algebra R B] [instCFC : ContinuousFunctionalCalculus R A p] (e : A ≃⋆ₐ[R] B) (hpq : ∀ (x : A), p x ↔ q (e x)) (b : B) (hb : q b) (x : C(↑(spectrum R b), R)) : (cfcHomTransfer e hpq b hb) x = e ((cfcHom ⋯) (x.comp ↑(Homeomorph.setCongr ⋯).symm))
IsCoveringMap.homotopicRel_iff_comp 📋 Mathlib.Topology.Homotopy.Lifting
{E : Type u_1} {X : Type u_2} {A : Type u_3} [TopologicalSpace E] [TopologicalSpace X] [TopologicalSpace A] {p : E → X} (cov : IsCoveringMap p) [PreconnectedSpace A] {f₀ f₁ : C(A, E)} {S : Set A} (he : ∃ a ∈ S, f₀ a = f₁ a) : f₀.HomotopicRel f₁ S ↔ ({ toFun := p, continuous_toFun := ⋯ }.comp f₀).HomotopicRel ({ toFun := p, continuous_toFun := ⋯ }.comp f₁) S
IsCoveringMap.liftHomotopy_apply 📋 Mathlib.Topology.Homotopy.Lifting
{E : Type u_1} {X : Type u_2} {A : Type u_3} [TopologicalSpace E] [TopologicalSpace X] [TopologicalSpace A] {p : E → X} (cov : IsCoveringMap p) (H : C(↑unitInterval × A, X)) (f : C(A, E)) (H_0 : ∀ (a : A), H (0, a) = p (f a)) (ta : ↑unitInterval × A) : (cov.liftHomotopy H f H_0) ta = (cov.liftPath (H.comp ((ContinuousMap.id ↑unitInterval).prodMk (ContinuousMap.const (↑unitInterval) ta.2))) (f ta.2) ⋯) ta.1
IsLocalHomeomorph.existsUnique_continuousMap_lifts 📋 Mathlib.Topology.Homotopy.Lifting
{E : Type u_1} {X : Type u_2} {A : Type u_3} [TopologicalSpace E] [TopologicalSpace X] [TopologicalSpace A] {p : E → X} (homeo : IsLocalHomeomorph p) [PathConnectedSpace A] [LocallyPathConnectedSpace A] (f : C(A, X)) (a₀ : A) (e₀ : E) (he : p e₀ = f a₀) (ex : ∀ (γ : C(↑unitInterval, A)), γ 0 = a₀ → ∃ Γ, Γ 0 = e₀ ∧ p ∘ ⇑Γ = ⇑(f.comp γ)) (uniq : ∀ (γ γ' : C(↑unitInterval, A)) (Γ Γ' : C(↑unitInterval, E)), γ 0 = a₀ → γ' 0 = a₀ → Γ 0 = e₀ → Γ' 0 = e₀ → p ∘ ⇑Γ = ⇑(f.comp γ) → p ∘ ⇑Γ' = ⇑(f.comp γ') → γ 1 = γ' 1 → Γ 1 = Γ' 1) : ∃! F, F a₀ = e₀ ∧ p ∘ ⇑F = ⇑f
TietzeExtension.of_retract 📋 Mathlib.Topology.TietzeExtension
{Y : Type v} {Z : Type w} [TopologicalSpace Y] [TopologicalSpace Z] [TietzeExtension Z] (ι : C(Y, Z)) (r : C(Z, Y)) (h : r.comp ι = ContinuousMap.id Y) : TietzeExtension Y
ContinuousMap.exists_extension 📋 Mathlib.Topology.TietzeExtension
{X₁ : Type u₁} [TopologicalSpace X₁] {X : Type u} [TopologicalSpace X] [NormalSpace X] {e : X₁ → X} {Y : Type v} [TopologicalSpace Y] [TietzeExtension Y] (he : Topology.IsClosedEmbedding e) (f : C(X₁, Y)) : ∃ g, g.comp { toFun := e, continuous_toFun := ⋯ } = f
ContinuousMap.exists_extension_forall_mem 📋 Mathlib.Topology.TietzeExtension
{X₁ : Type u₁} [TopologicalSpace X₁] {X : Type u} [TopologicalSpace X] [NormalSpace X] {e : X₁ → X} (he : Topology.IsClosedEmbedding e) {Y : Type v} [TopologicalSpace Y] (f : C(X₁, Y)) {t : Set Y} (hf : ∀ (x : X₁), f x ∈ t) [ht : TietzeExtension ↑t] : ∃ g, (∀ (x : X), g x ∈ t) ∧ g.comp { toFun := e, continuous_toFun := ⋯ } = f
CompHausLike.ofHom_comp 📋 Mathlib.Topology.Category.CompHausLike.Basic
(P : TopCat → Prop) {X : Type u} [TopologicalSpace X] [CompactSpace X] [T2Space X] [CompHausLike.HasProp P X] {Y : Type u} [TopologicalSpace Y] [CompactSpace Y] [T2Space Y] [CompHausLike.HasProp P Y] {Z : Type u} [TopologicalSpace Z] [CompactSpace Z] [T2Space Z] [CompHausLike.HasProp P Z] (f : C(X, Y)) (g : C(Y, Z)) : CompHausLike.ofHom P (g.comp f) = CategoryTheory.CategoryStruct.comp (CompHausLike.ofHom P f) (CompHausLike.ofHom P g)
Real.integrable_of_summable_norm_Icc 📋 Mathlib.MeasureTheory.Measure.Lebesgue.Integral
{E : Type u_1} [NormedAddCommGroup E] {f : C(ℝ, E)} (hf : Summable fun n => ‖ContinuousMap.restrict (Set.Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))‖) : MeasureTheory.Integrable (⇑f) MeasureTheory.volume
ContinuousMap.periodic_tsum_comp_add_zsmul 📋 Mathlib.Topology.ContinuousMap.Periodic
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [AddCommGroup X] [ContinuousAdd X] [AddCommMonoid Y] [ContinuousAdd Y] [T2Space Y] (f : C(X, Y)) (p : X) : Function.Periodic (⇑(∑' (n : ℤ), f.comp (ContinuousMap.addRight (n • p)))) p
isBigO_norm_restrict_cocompact 📋 Mathlib.Analysis.Fourier.PoissonSummation
{E : Type u_1} [NormedAddCommGroup E] (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b) (hf : ⇑f =O[Filter.cocompact ℝ] fun x => |x| ^ (-b)) (K : TopologicalSpace.Compacts ℝ) : (fun x => ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))‖) =O[Filter.cocompact ℝ] fun x => |x| ^ (-b)
Real.tsum_eq_tsum_fourier 📋 Mathlib.Analysis.Fourier.PoissonSummation
{f : C(ℝ, ℂ)} (h_norm : ∀ (K : TopologicalSpace.Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖) (h_sum : Summable fun n => FourierTransform.fourier ⇑f ↑n) (x : ℝ) : ∑' (n : ℤ), f (x + ↑n) = ∑' (n : ℤ), FourierTransform.fourier ⇑f ↑n * (fourier n) ↑x
Real.fourierCoeff_tsum_comp_add 📋 Mathlib.Analysis.Fourier.PoissonSummation
{f : C(ℝ, ℂ)} (hf : ∀ (K : TopologicalSpace.Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖) (m : ℤ) : fourierCoeff ⋯.lift m = FourierTransform.fourier ⇑f ↑m
DiscreteQuotient.comap_comp 📋 Mathlib.Topology.DiscreteQuotient
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (g : C(Y, Z)) (f : C(X, Y)) (S : DiscreteQuotient Z) : DiscreteQuotient.comap (g.comp f) S = DiscreteQuotient.comap f (DiscreteQuotient.comap g S)
DiscreteQuotient.LEComap.comp 📋 Mathlib.Topology.DiscreteQuotient
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} {A : DiscreteQuotient X} {B : DiscreteQuotient Y} {g : C(Y, Z)} {C : DiscreteQuotient Z} : DiscreteQuotient.LEComap g B C → DiscreteQuotient.LEComap f A B → DiscreteQuotient.LEComap (g.comp f) A C
DiscreteQuotient.map_comp 📋 Mathlib.Topology.DiscreteQuotient
{X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} {A : DiscreteQuotient X} {B : DiscreteQuotient Y} {g : C(Y, Z)} {C : DiscreteQuotient Z} (h1 : DiscreteQuotient.LEComap g B C) (h2 : DiscreteQuotient.LEComap f A B) : DiscreteQuotient.map (g.comp f) ⋯ = DiscreteQuotient.map g h1 ∘ DiscreteQuotient.map f h2
ContinuousMap.yonedaPresheaf'_map 📋 Mathlib.Topology.Category.TopCat.Yoneda
(Y : Type w') [TopologicalSpace Y] {X✝ Y✝ : TopCatᵒᵖ} (f : X✝ ⟶ Y✝) : (ContinuousMap.yonedaPresheaf' Y).map f = TypeCat.ofHom fun g => g.comp (CategoryTheory.ConcreteCategory.hom f.unop)
ContinuousMap.yonedaPresheaf_map 📋 Mathlib.Topology.Category.TopCat.Yoneda
{C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor C TopCat) (Y : Type w') [TopologicalSpace Y] {X✝ Y✝ : Cᵒᵖ} (f : X✝ ⟶ Y✝) : (ContinuousMap.yonedaPresheaf F Y).map f = TypeCat.ofHom fun g => g.comp (TopCat.Hom.hom (F.map f.unop))
TopCat.toSheafCompHausLike_obj_map 📋 Mathlib.Condensed.TopComparison
(P : TopCat → Prop) (X : TopCat) [CompHausLike.HasExplicitFiniteCoproducts P] [CompHausLike.HasExplicitPullbacks P] (hs : ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y), CategoryTheory.EffectiveEpi f → Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom f)) {X✝ Y✝ : (CompHausLike P)ᵒᵖ} (f : X✝ ⟶ Y✝) : (TopCat.toSheafCompHausLike P X hs).obj.map f = TypeCat.ofHom fun g => g.comp (TopCat.Hom.hom f.unop.hom)
topCatToSheafCompHausLike_map_hom_app 📋 Mathlib.Condensed.TopComparison
(P : TopCat → Prop) [CompHausLike.HasExplicitFiniteCoproducts P] [CompHausLike.HasExplicitPullbacks P] (hs : ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y), CategoryTheory.EffectiveEpi f → Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom f)) {X✝ Y✝ : TopCat} (f : X✝ ⟶ Y✝) (x✝ : (CompHausLike P)ᵒᵖ) : ((topCatToSheafCompHausLike P hs).map f).hom.app x✝ = TypeCat.ofHom fun g => (TopCat.Hom.hom f).comp g
CompactlySupportedContinuousMap.toContinuousMap_compLeft 📋 Mathlib.Topology.ContinuousMap.CompactlySupported
{α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {γ : Type u_5} [TopologicalSpace γ] [Zero γ] {g : C(β, γ)} (hg : g 0 = 0) (f : CompactlySupportedContinuousMap α β) : (CompactlySupportedContinuousMap.compLeft g f).toContinuousMap = g.comp ↑f
ContinuousMap.prodMul_def 📋 Mathlib.Topology.UniformSpace.ProdApproximation
{X : Type u_5} {Y : Type u_6} {R : Type u_7} [TopologicalSpace X] [TopologicalSpace Y] [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] (f : C(X, R)) (g : C(Y, R)) : (ContinuousMap.prodMul f) g = f.comp ContinuousMap.fst * g.comp ContinuousMap.snd
AbstractMeasure.map_apply 📋 Mathlib.NumberTheory.Padics.Measure.Basic
{X : Type u_1} {Y : Type u_2} {R : Type u_3} {E : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [AddCommGroup E] [TopologicalSpace E] [IsTopologicalAddGroup E] [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] [Module R E] [ContinuousSMul R E] (f : C(X, Y)) (μ : AbstractMeasure X R E) (g : C(Y, R)) : ((AbstractMeasure.map f) μ) g = μ (g.comp f)
AbstractMeasure.arrowCongrLeft_apply 📋 Mathlib.NumberTheory.Padics.Measure.Basic
{X : Type u_1} {Y : Type u_2} {R : Type u_3} {E : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [AddCommGroup E] [TopologicalSpace E] [IsTopologicalAddGroup E] [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] [Module R E] [ContinuousSMul R E] (φ : X ≃ₜ Y) (μ : AbstractMeasure X R E) (f : C(Y, R)) : ((AbstractMeasure.arrowCongrLeft φ) μ) f = μ (f.comp ↑φ)
AbstractMeasure.map_map 📋 Mathlib.NumberTheory.Padics.Measure.Basic
{X : Type u_1} {Y : Type u_2} {R : Type u_3} {E : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [AddCommGroup E] [TopologicalSpace E] [IsTopologicalAddGroup E] [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] [Module R E] [ContinuousSMul R E] {Z : Type u_5} [TopologicalSpace Z] (f : C(X, Y)) (g : C(Y, Z)) (μ : AbstractMeasure X R E) : (AbstractMeasure.map g) ((AbstractMeasure.map f) μ) = (AbstractMeasure.map (g.comp f)) μ
AbstractMeasure.prodMk'_prod_apply 📋 Mathlib.NumberTheory.Padics.Measure.Basic
{X : Type u_1} {Y : Type u_2} {R : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] (μ : AbstractMeasure X R R) (ν : AbstractMeasure Y R R) [LocallyCompactSpace X] [LocallyCompactSpace Y] (f : C(X, R)) (g : C(Y, R)) : ((AbstractMeasure.prodMk' μ) ν) (f.comp ContinuousMap.fst * g.comp ContinuousMap.snd) = μ f * ν g
AbstractMeasure.prodMk_prod_apply 📋 Mathlib.NumberTheory.Padics.Measure.Basic
{X : Type u_1} {Y : Type u_2} {R : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] (μ : AbstractMeasure X R R) (ν : AbstractMeasure Y R R) [LocallyCompactSpace X] [LocallyCompactSpace Y] (f : C(X, R)) (g : C(Y, R)) : ((AbstractMeasure.prodMk μ) ν) (f.comp ContinuousMap.fst * g.comp ContinuousMap.snd) = μ f * ν g
AbstractMeasure.prodMk'_flip 📋 Mathlib.NumberTheory.Padics.Measure.Basic
{X : Type u_1} {Y : Type u_2} {R : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] (μ : AbstractMeasure X R R) (ν : AbstractMeasure Y R R) [LocallyCompactSpace X] [LocallyCompactSpace Y] (f : C(X × Y, R)) : ((AbstractMeasure.prodMk' μ) ν) f = ((AbstractMeasure.prodMk ν) μ) (f.comp ContinuousMap.prodSwap)
ContRepresentation.coind₁Map_toFun 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} {W : Type u_4} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] [AddCommGroup W] [TopologicalSpace W] [IsTopologicalAddGroup W] [Module R W] {G : Type u_7} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [ContinuousSMul R W] [IsTopologicalGroup G] {π₁ : ContRepresentation R G V} {π₂ : ContRepresentation R G W} (f : ContIntertwiningMap π₁ π₂) (g : C(G, V)) : (ContRepresentation.coind₁Map f) g = (↑f).comp g
ContRepresentation.coind_toMonoidHom_apply_apply_coe 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [Group H] [TopologicalSpace H] (φ : G →ₜ* H) [IsTopologicalGroup H] (π : ContRepresentation R G V) (h : H) (x✝ : ↥(ContRepresentation.coindV φ π)) : ↑(((ContRepresentation.coind φ π).toMonoidHom h) x✝) = (↑x✝).comp (ContinuousMap.mulRight h)
ContinuousMap.concat_comp_IccInclusionLeft 📋 Mathlib.Topology.ContinuousMap.Interval
{α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] {a b c : α} [Fact (a ≤ b)] [Fact (b ≤ c)] {E : Type u_2} [TopologicalSpace E] {f : C(↑(Set.Icc a b), E)} {g : C(↑(Set.Icc b c), E)} (hb : f ⊤ = g ⊥) : (f.concat g).comp ContinuousMap.IccInclusionLeft = f
ContinuousMap.concat_comp_IccInclusionRight 📋 Mathlib.Topology.ContinuousMap.Interval
{α : Type u_1} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] {a b c : α} [Fact (a ≤ b)] [Fact (b ≤ c)] {E : Type u_2} [TopologicalSpace E] {f : C(↑(Set.Icc a b), E)} {g : C(↑(Set.Icc b c), E)} (hb : f ⊤ = g ⊥) : (f.concat g).comp ContinuousMap.IccInclusionRight = g
ContinuousMap.exists_lift_sigma 📋 Mathlib.Topology.ContinuousMap.Sigma
{X : Type u_1} {ι : Type u_2} {Y : ι → Type u_3} [TopologicalSpace X] [(i : ι) → TopologicalSpace (Y i)] [ConnectedSpace X] (f : C(X, (i : ι) × Y i)) : ∃ i g, f = (ContinuousMap.sigmaMk i).comp g
ContinuousMap.isEmbedding_sigmaMk_comp 📋 Mathlib.Topology.ContinuousMap.Sigma
{X : Type u_1} {ι : Type u_2} {Y : ι → Type u_3} [TopologicalSpace X] [(i : ι) → TopologicalSpace (Y i)] [Nonempty X] : Topology.IsEmbedding fun g => (ContinuousMap.sigmaMk g.fst).comp g.snd
ContinuousMap.sigmaCodHomeomorph_symm_apply 📋 Mathlib.Topology.ContinuousMap.Sigma
(X : Type u_1) {ι : Type u_2} (Y : ι → Type u_3) [TopologicalSpace X] [(i : ι) → TopologicalSpace (Y i)] [ConnectedSpace X] (a✝ : (i : ι) × C(X, Y i)) : (ContinuousMap.sigmaCodHomeomorph X Y).symm a✝ = (ContinuousMap.sigmaMk a✝.fst).comp a✝.snd
HSpace.eHmul 📋 Mathlib.Topology.Homotopy.HSpaces
{X : Type u} {inst✝ : TopologicalSpace X} [self : HSpace X] : (HSpace.hmul.comp ((ContinuousMap.const X HSpace.e).prodMk (ContinuousMap.id X))).HomotopyRel (ContinuousMap.id X) {HSpace.e}
HSpace.hmulE 📋 Mathlib.Topology.Homotopy.HSpaces
{X : Type u} {inst✝ : TopologicalSpace X} [self : HSpace X] : (HSpace.hmul.comp ((ContinuousMap.id X).prodMk (ContinuousMap.const X HSpace.e))).HomotopyRel (ContinuousMap.id X) {HSpace.e}
HSpace.mk 📋 Mathlib.Topology.Homotopy.HSpaces
{X : Type u} [TopologicalSpace X] (hmul : C(X × X, X)) (e : X) (hmul_e_e : hmul (e, e) = e) (eHmul : (hmul.comp ((ContinuousMap.const X e).prodMk (ContinuousMap.id X))).HomotopyRel (ContinuousMap.id X) {e}) (hmulE : (hmul.comp ((ContinuousMap.id X).prodMk (ContinuousMap.const X e))).HomotopyRel (ContinuousMap.id X) {e}) : HSpace X
GenLoop.currySum_apply_coe 📋 Mathlib.Topology.Homotopy.HomotopyGroup
{N : Type u_1} {X : Type u_2} [TopologicalSpace X] {M : Type u_3} (x : X) (q : ↑(GenLoop (M ⊕ N) X x)) (a : M → ↑unitInterval) : ↑((GenLoop.currySum x q) a) = ((↑q).comp { toFun := Homeomorph.sumArrowHomeomorphProdArrow.invFun, continuous_toFun := ⋯ }).curry.toFun a
GenLoop.toLoop_apply_coe 📋 Mathlib.Topology.Homotopy.HomotopyGroup
{N : Type u_1} {X : Type u_2} [TopologicalSpace X] {x : X} [DecidableEq N] (i : N) (p : ↑(GenLoop N X x)) (t : ↑unitInterval) : ↑((GenLoop.toLoop i p) t) = ((↑p).comp ↑(Cube.insertAt i)).curry t
GenLoop.fromLoop_coe 📋 Mathlib.Topology.Homotopy.HomotopyGroup
{N : Type u_1} {X : Type u_2} [TopologicalSpace X] {x : X} [DecidableEq N] (i : N) (p : LoopSpace (↑(GenLoop { j // j ≠ i } X x)) GenLoop.const) : ↑(GenLoop.fromLoop i p) = ({ toFun := Subtype.val, continuous_toFun := ⋯ }.comp p.toContinuousMap).uncurry.comp ↑(Cube.splitAt i)
GenLoop.genLoopGenLoopEquiv_apply_coe 📋 Mathlib.Topology.Homotopy.HomotopyGroup
{N : Type u_1} {X : Type u_2} [TopologicalSpace X] {M : Type u_3} (x : X) (p : ↑(GenLoop M (↑(GenLoop N X x)) GenLoop.const)) : ↑((GenLoop.genLoopGenLoopEquiv x) p) = (GenLoop.uncurry x p).comp { toFun := Homeomorph.sumArrowHomeomorphProdArrow.toFun, continuous_toFun := ⋯ }
Topology.WithLowerSet.map_comp 📋 Mathlib.Topology.Order.UpperLowerSetTopology
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [Preorder α] [Preorder β] [Preorder γ] (g : β →o γ) (f : α →o β) : Topology.WithLowerSet.map (g.comp f) = (Topology.WithLowerSet.map g).comp (Topology.WithLowerSet.map f)
Topology.WithUpperSet.map_comp 📋 Mathlib.Topology.Order.UpperLowerSetTopology
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [Preorder α] [Preorder β] [Preorder γ] (g : β →o γ) (f : α →o β) : Topology.WithUpperSet.map (g.comp f) = (Topology.WithUpperSet.map g).comp (Topology.WithUpperSet.map f)
Specialization.map_comp 📋 Mathlib.Topology.Specialization
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (g : C(β, γ)) (f : C(α, β)) : Specialization.map (g.comp f) = (Specialization.map g).comp (Specialization.map f)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c