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Found 122 declarations mentioning IsTopologicalAddTorsor.

IsTopologicalAddTorsor 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} [AddGroup V] [TopologicalSpace V] (P : Type u_2) [AddTorsor V P] [TopologicalSpace P] : Prop
instIsTopologicalAddTorsor 📋 Mathlib.Topology.Algebra.Group.Torsor
{G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] : IsTopologicalAddTorsor G
IsTopologicalAddTorsor.regularSpace 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] : RegularSpace P
IsTopologicalAddTorsor.to_isTopologicalAddGroup 📋 Mathlib.Topology.Algebra.Group.Torsor
(V : Type u_1) (P : Type u_2) [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] : IsTopologicalAddGroup V
Homeomorph.constVSub 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] (p : P) : P ≃ₜ V
Homeomorph.pointReflection 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_4} {P : Type u_5} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] (p : P) : P ≃ₜ P
Homeomorph.vaddConst 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] (p : P) : V ≃ₜ P
IsTopologicalAddTorsor.t0Space 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [T0Space V] : T0Space P
IsTopologicalAddTorsor.toContinuousVAdd 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {inst✝ : AddGroup V} {inst✝¹ : TopologicalSpace V} {P : Type u_2} {inst✝² : AddTorsor V P} {inst✝³ : TopologicalSpace P} [self : IsTopologicalAddTorsor P] : ContinuousVAdd V P
IsTopologicalAddTorsor.continuous_vsub 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {inst✝ : AddGroup V} {inst✝¹ : TopologicalSpace V} {P : Type u_2} {inst✝² : AddTorsor V P} {inst✝³ : TopologicalSpace P} [self : IsTopologicalAddTorsor P] : Continuous fun x => x.1 -ᵥ x.2
Continuous.vsub 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} {α : Type u_3} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [TopologicalSpace α] {f g : α → P} (hf : Continuous f) (hg : Continuous g) : Continuous fun x => f x -ᵥ g x
ContinuousAt.vsub 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} {α : Type u_3} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [TopologicalSpace α] {f g : α → P} {x : α} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun x => f x -ᵥ g x) x
ContinuousOn.vsub 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} {α : Type u_3} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [TopologicalSpace α] {f g : α → P} {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) : ContinuousOn (fun x => f x -ᵥ g x) s
ContinuousWithinAt.vsub 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} {α : Type u_3} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [TopologicalSpace α] {f g : α → P} {x : α} {s : Set α} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) : ContinuousWithinAt (fun x => f x -ᵥ g x) s x
Homeomorph.constVSub_apply 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] (p x✝ : P) : (Homeomorph.constVSub p) x✝ = p -ᵥ x✝
Homeomorph.vaddConst_symm_apply 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] (p p' : P) : (Homeomorph.vaddConst p).symm p' = p' -ᵥ p
Homeomorph.coe_pointReflection 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_4} {P : Type u_5} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] (p : P) : ⇑(Homeomorph.pointReflection p) = ⇑(Equiv.pointReflection p)
IsTopologicalAddTorsor.mk 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} [AddGroup V] [TopologicalSpace V] {P : Type u_2} [AddTorsor V P] [TopologicalSpace P] [toContinuousVAdd : ContinuousVAdd V P] (continuous_vsub : Continuous fun x => x.1 -ᵥ x.2) : IsTopologicalAddTorsor P
instIsTopologicalAddTorsorForall 📋 Mathlib.Topology.Algebra.Group.Torsor
{ι : Type u_1} {V : ι → Type u_2} {P : ι → Type u_3} [(i : ι) → AddCommGroup (V i)] [(i : ι) → TopologicalSpace (V i)] [(i : ι) → AddTorsor (V i) (P i)] [(i : ι) → TopologicalSpace (P i)] [∀ (i : ι), IsTopologicalAddTorsor (P i)] : IsTopologicalAddTorsor ((i : ι) → P i)
instIsTopologicalAddTorsorProd 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {W : Type u_2} {P : Type u_3} {Q : Type u_4} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [AddCommGroup W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] : IsTopologicalAddTorsor (P × Q)
Homeomorph.vaddConst_apply 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] (p : P) (v : V) : (Homeomorph.vaddConst p) v = v +ᵥ p
Filter.Tendsto.vsub 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} {α : Type u_3} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] {l : Filter α} {f g : α → P} {x y : P} (hf : Filter.Tendsto f l (nhds x)) (hg : Filter.Tendsto g l (nhds y)) : Filter.Tendsto (f -ᵥ g) l (nhds (x -ᵥ y))
Homeomorph.constVSub_symm_apply 📋 Mathlib.Topology.Algebra.Group.Torsor
{V : Type u_1} {P : Type u_2} [AddGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] (p : P) (x✝ : V) : (Homeomorph.constVSub p).symm x✝ = -x✝ +ᵥ p
AffineMap.homothety_continuous 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [CommRing R] [Module R V] [ContinuousConstSMul R V] (x : P) (t : R) : Continuous ⇑(AffineMap.homothety x t)
AffineMap.lineMap_continuous 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [Ring R] [Module R V] [TopologicalSpace R] [ContinuousSMul R V] {p q : P} : Continuous ⇑(AffineMap.lineMap p q)
AffineMap.homothety_isOpenMap 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [Field R] [Module R V] [ContinuousConstSMul R V] (x : P) (t : R) (ht : t ≠ 0) : IsOpenMap ⇑(AffineMap.homothety x t)
eventually_homothety_mem_of_mem_interior 📋 Mathlib.Topology.Algebra.Affine
(R : Type u_1) {W : Type u_4} {Q : Type u_5} [AddCommGroup W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] [CommRing R] [TopologicalSpace R] [Module R W] [ContinuousSMul R W] (x : Q) {s : Set Q} {y : Q} (hy : y ∈ interior s) : ∀ᶠ (δ : R) in nhds 1, (AffineMap.homothety x δ) y ∈ s
AffineMap.continuous_linear_iff 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} {W : Type u_4} {Q : Type u_5} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [AddCommGroup W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] [Ring R] [Module R V] [Module R W] {f : P →ᵃ[R] Q} : Continuous ⇑f.linear ↔ Continuous ⇑f
AffineMap.isClosedEmbedding_linear_iff 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} {W : Type u_4} {Q : Type u_5} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [AddCommGroup W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] [Ring R] [Module R V] [Module R W] {f : P →ᵃ[R] Q} : Topology.IsClosedEmbedding ⇑f.linear ↔ Topology.IsClosedEmbedding ⇑f
AffineMap.isEmbedding_linear_iff 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} {W : Type u_4} {Q : Type u_5} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [AddCommGroup W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] [Ring R] [Module R V] [Module R W] {f : P →ᵃ[R] Q} : Topology.IsEmbedding ⇑f.linear ↔ Topology.IsEmbedding ⇑f
AffineMap.isOpenEmbedding_linear_iff 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} {W : Type u_4} {Q : Type u_5} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [AddCommGroup W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] [Ring R] [Module R V] [Module R W] {f : P →ᵃ[R] Q} : Topology.IsOpenEmbedding ⇑f.linear ↔ Topology.IsOpenEmbedding ⇑f
AffineMap.isOpenMap_linear_iff 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} {W : Type u_4} {Q : Type u_5} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [AddCommGroup W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] [Ring R] [Module R V] [Module R W] {f : P →ᵃ[R] Q} : IsOpenMap ⇑f.linear ↔ IsOpenMap ⇑f
eventually_homothety_image_subset_of_finite_subset_interior 📋 Mathlib.Topology.Algebra.Affine
(R : Type u_1) {W : Type u_4} {Q : Type u_5} [AddCommGroup W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] [CommRing R] [TopologicalSpace R] [Module R W] [ContinuousSMul R W] (x : Q) {s t : Set Q} (ht : t.Finite) (h : t ⊆ interior s) : ∀ᶠ (δ : R) in nhds 1, ⇑(AffineMap.homothety x δ) '' t ⊆ s
Filter.Tendsto.midpoint 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [Ring R] [Module R V] [TopologicalSpace R] [ContinuousSMul R V] {α : Type u_6} {l : Filter α} [Invertible 2] {f₁ f₂ : α → P} {p₁ p₂ : P} (h₁ : Filter.Tendsto f₁ l (nhds p₁)) (h₂ : Filter.Tendsto f₂ l (nhds p₂)) : Filter.Tendsto (fun x => midpoint R (f₁ x) (f₂ x)) l (nhds (midpoint R p₁ p₂))
Continuous.lineMap 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [Ring R] [Module R V] [TopologicalSpace R] [ContinuousSMul R V] {X : Type u_6} [TopologicalSpace X] {f₁ f₂ : X → P} {g : X → R} (h₁ : Continuous f₁) (h₂ : Continuous f₂) (hg : Continuous g) : Continuous fun x => (AffineMap.lineMap (f₁ x) (f₂ x)) (g x)
ContinuousAt.lineMap 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [Ring R] [Module R V] [TopologicalSpace R] [ContinuousSMul R V] {X : Type u_6} [TopologicalSpace X] {f₁ f₂ : X → P} {g : X → R} {x : X} (h₁ : ContinuousAt f₁ x) (h₂ : ContinuousAt f₂ x) (hg : ContinuousAt g x) : ContinuousAt (fun x => (AffineMap.lineMap (f₁ x) (f₂ x)) (g x)) x
ContinuousOn.lineMap 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [Ring R] [Module R V] [TopologicalSpace R] [ContinuousSMul R V] {X : Type u_6} [TopologicalSpace X] {f₁ f₂ : X → P} {g : X → R} {s : Set X} (h₁ : ContinuousOn f₁ s) (h₂ : ContinuousOn f₂ s) (hg : ContinuousOn g s) : ContinuousOn (fun x => (AffineMap.lineMap (f₁ x) (f₂ x)) (g x)) s
AffineMap.lineMap_continuous_uncurry 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [Ring R] [Module R V] [TopologicalSpace R] [ContinuousSMul R V] : Continuous fun pqt => (AffineMap.lineMap pqt.1 pqt.2.1) pqt.2.2
ContinuousWithinAt.lineMap 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [Ring R] [Module R V] [TopologicalSpace R] [ContinuousSMul R V] {X : Type u_6} [TopologicalSpace X] {f₁ f₂ : X → P} {g : X → R} {s : Set X} {x : X} (h₁ : ContinuousWithinAt f₁ s x) (h₂ : ContinuousWithinAt f₂ s x) (hg : ContinuousWithinAt g s x) : ContinuousWithinAt (fun x => (AffineMap.lineMap (f₁ x) (f₂ x)) (g x)) s x
Filter.Tendsto.lineMap 📋 Mathlib.Topology.Algebra.Affine
{R : Type u_1} {V : Type u_2} {P : Type u_3} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [Ring R] [Module R V] [TopologicalSpace R] [ContinuousSMul R V] {α : Type u_6} {l : Filter α} {f₁ f₂ : α → P} {g : α → R} {p₁ p₂ : P} {c : R} (h₁ : Filter.Tendsto f₁ l (nhds p₁)) (h₂ : Filter.Tendsto f₂ l (nhds p₂)) (hg : Filter.Tendsto g l (nhds c)) : Filter.Tendsto (fun x => (AffineMap.lineMap (f₁ x) (f₂ x)) (g x)) l (nhds ((AffineMap.lineMap p₁ p₂) c))
Affine.Simplex.isCompact_closedInterior 📋 Mathlib.Analysis.Convex.Topology
{𝕜 : Type u_3} {V : Type u_4} {P : Type u_5} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜] [OrderClosedTopology 𝕜] [CompactIccSpace 𝕜] [IsTopologicalRing 𝕜] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module 𝕜 V] [ContinuousSMul 𝕜 V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] {n : ℕ} (s : Affine.Simplex 𝕜 P n) : IsCompact s.closedInterior
Affine.Simplex.isClosed_closedInterior 📋 Mathlib.Analysis.Convex.Topology
{𝕜 : Type u_3} {V : Type u_4} {P : Type u_5} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜] [OrderClosedTopology 𝕜] [CompactIccSpace 𝕜] [IsTopologicalRing 𝕜] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module 𝕜 V] [ContinuousSMul 𝕜 V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [T2Space P] {n : ℕ} (s : Affine.Simplex 𝕜 P n) : IsClosed s.closedInterior
instIsTopologicalAddTorsor_1 📋 Mathlib.Analysis.Normed.Group.AddTorsor
{V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] : IsTopologicalAddTorsor P
ContinuousAffineMap.decompEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
(R : Type u_1) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] : (V →ᴬ[R] Q) ≃ Q × (V →L[R] W)
ContinuousAffineMap.contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace V] [IsTopologicalAddTorsor P] [TopologicalSpace W] [IsTopologicalAddTorsor Q] (f : P →ᴬ[R] Q) : V →L[R] W
ContinuousAffineMap.instAddTorsor 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace W] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] : AddTorsor (P →ᴬ[R] W) (P →ᴬ[R] Q)
ContinuousAffineMap.coe_contLinear_eq_linear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace V] [IsTopologicalAddTorsor P] [TopologicalSpace W] [IsTopologicalAddTorsor Q] (f : P →ᴬ[R] Q) : ↑f.contLinear = (↑f).linear
ContinuousAffineMap.const_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace V] [IsTopologicalAddTorsor P] [TopologicalSpace W] [IsTopologicalAddTorsor Q] (q : Q) : (ContinuousAffineMap.const R P q).contLinear = 0
ContinuousAffineMap.neg_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [IsTopologicalAddGroup W] [TopologicalSpace V] [IsTopologicalAddTorsor P] (f : P →ᴬ[R] W) : (-f).contLinear = -f.contLinear
ContinuousAffineMap.coe_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace V] [IsTopologicalAddTorsor P] [TopologicalSpace W] [IsTopologicalAddTorsor Q] (f : P →ᴬ[R] Q) : ⇑f.contLinear = ⇑(↑f).linear
ContinuousAffineMap.coe_linear_eq_coe_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace V] [IsTopologicalAddTorsor P] [TopologicalSpace W] [IsTopologicalAddTorsor Q] (f : P →ᴬ[R] Q) : ⇑(↑f).linear = ⇑f.contLinear
ContinuousAffineMap.zero_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [IsTopologicalAddGroup W] [TopologicalSpace V] [IsTopologicalAddTorsor P] : ContinuousAffineMap.contLinear 0 = 0
ContinuousAffineMap.contLinear_eq_zero_iff_exists_const 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace V] [IsTopologicalAddTorsor P] [TopologicalSpace W] [IsTopologicalAddTorsor Q] (f : P →ᴬ[R] Q) : f.contLinear = 0 ↔ ∃ q, f = ContinuousAffineMap.const R P q
ContinuousAffineMap.contLinear_map_vsub 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace V] [IsTopologicalAddTorsor P] [TopologicalSpace W] [IsTopologicalAddTorsor Q] (f : P →ᴬ[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂
ContinuousAffineMap.coe_mk_contLinear_eq_linear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace V] [IsTopologicalAddTorsor P] [TopologicalSpace W] [IsTopologicalAddTorsor Q] (f : P →ᵃ[R] Q) (h : Continuous f.toFun) : ⇑{ toAffineMap := f, cont := h }.contLinear = ⇑f.linear
ContinuousAffineMap.comp_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] {W₂ : Type u_6} {Q₂ : Type u_7} [AddCommGroup W₂] [Module R W₂] [TopologicalSpace Q₂] [AddTorsor W₂ Q₂] [TopologicalSpace V] [IsTopologicalAddTorsor P] [TopologicalSpace W] [IsTopologicalAddTorsor Q] [TopologicalSpace W₂] [IsTopologicalAddTorsor Q₂] (f : P →ᴬ[R] Q) (g : Q →ᴬ[R] Q₂) : (g.comp f).contLinear = g.contLinear ∘SL f.contLinear
ContinuousAffineMap.map_vadd 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace V] [IsTopologicalAddTorsor P] [TopologicalSpace W] [IsTopologicalAddTorsor Q] (f : P →ᴬ[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p
ContinuousAffineMap.vsub_apply 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace W] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (f g : P →ᴬ[R] Q) (p : P) : (f -ᵥ g) p = f p -ᵥ g p
ContinuousAffineMap.prod_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{k : Type u_8} {P₁ : Type u_9} {P₂ : Type u_10} {P₃ : Type u_11} {V₁ : Type u_13} {V₂ : Type u_14} {V₃ : Type u_15} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₂] [AddCommGroup V₃] [Module k V₃] [AddTorsor V₃ P₃] [TopologicalSpace P₃] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] [TopologicalSpace V₂] [IsTopologicalAddTorsor P₂] [TopologicalSpace V₃] [IsTopologicalAddTorsor P₃] (f : P₁ →ᴬ[k] P₂) (g : P₁ →ᴬ[k] P₃) : (f.prod g).contLinear = f.contLinear.prod g.contLinear
ContinuousAffineMap.vsub_toAffineMap 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace W] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (f g : P →ᴬ[R] Q) : ↑(f -ᵥ g) = ↑f -ᵥ ↑g
ContinuousAffineMap.sub_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [IsTopologicalAddGroup W] [TopologicalSpace V] [IsTopologicalAddTorsor P] (f g : P →ᴬ[R] W) : (f - g).contLinear = f.contLinear - g.contLinear
ContinuousAffineMap.snd_decompEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
(R : Type u_1) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] (f : V →ᴬ[R] Q) : ((ContinuousAffineMap.decompEquiv R V Q) f).2 = f.contLinear
ContinuousAffineMap.fst_decompEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
(R : Type u_1) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] (f : V →ᴬ[R] Q) : ((ContinuousAffineMap.decompEquiv R V Q) f).1 = f 0
ContinuousAffineMap.add_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [IsTopologicalAddGroup W] [TopologicalSpace V] [IsTopologicalAddTorsor P] (f g : P →ᴬ[R] W) : (f + g).contLinear = f.contLinear + g.contLinear
ContinuousAffineMap.vsub_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace W] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] [TopologicalSpace V] [IsTopologicalAddTorsor P] (f g : P →ᴬ[R] Q) : (f -ᵥ g).contLinear = f.contLinear - g.contLinear
ContinuousAffineMap.prodMap_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{k : Type u_8} {P₁ : Type u_9} {P₂ : Type u_10} {P₃ : Type u_11} {P₄ : Type u_12} {V₁ : Type u_13} {V₂ : Type u_14} {V₃ : Type u_15} {V₄ : Type u_16} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₂] [AddCommGroup V₃] [Module k V₃] [AddTorsor V₃ P₃] [TopologicalSpace P₃] [AddCommGroup V₄] [Module k V₄] [AddTorsor V₄ P₄] [TopologicalSpace P₄] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] [TopologicalSpace V₂] [IsTopologicalAddTorsor P₂] [TopologicalSpace V₃] [IsTopologicalAddTorsor P₃] [TopologicalSpace V₄] [IsTopologicalAddTorsor P₄] (f : P₁ →ᴬ[k] P₂) (g : P₃ →ᴬ[k] P₄) : (f.prodMap g).contLinear = f.contLinear.prodMap g.contLinear
ContinuousAffineMap.decompEquiv_symm_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
(R : Type u_1) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] (p : Q × (V →L[R] W)) : ((ContinuousAffineMap.decompEquiv R V Q).symm p).contLinear = p.2
ContinuousAffineMap.smul_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] {S : Type u_8} [TopologicalSpace W] [Monoid S] [DistribMulAction S W] [SMulCommClass R S W] [ContinuousConstSMul S W] [TopologicalSpace V] [IsTopologicalAddTorsor P] [IsTopologicalAddGroup W] (t : S) (f : P →ᴬ[R] W) : (t • f).contLinear = t • f.contLinear
ContinuousAffineMap.decompEquiv_symm_apply 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
(R : Type u_1) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] (p : Q × (V →L[R] W)) (x : V) : ((ContinuousAffineMap.decompEquiv R V Q).symm p) x = p.2 x +ᵥ p.1
ContinuousAffineMap.vadd_apply 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace W] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (f : P →ᴬ[R] W) (g : P →ᴬ[R] Q) (p : P) : (f +ᵥ g) p = f p +ᵥ g p
ContinuousAffineMap.lineMap_apply' 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace W] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] [ContinuousConstSMul R W] [SMulCommClass R R W] (f g : P →ᴬ[R] Q) (c : R) (p : P) : ((AffineMap.lineMap f g) c) p = (AffineMap.lineMap (f p) (g p)) c
ContinuousAffineMap.vadd_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace W] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] [TopologicalSpace V] [IsTopologicalAddTorsor P] (f : P →ᴬ[R] W) (g : P →ᴬ[R] Q) : (f +ᵥ g).contLinear = f.contLinear + g.contLinear
ContinuousAffineMap.decompAffineEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
(R : Type u_1) (S : Type u_2) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring S] [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [Module S W] [SMulCommClass R S W] [ContinuousConstSMul S W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] : (V →ᴬ[R] Q) ≃ᵃ[S] Q × (V →L[R] W)
ContinuousAffineMap.vadd_toAffineMap 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [AddCommGroup W] [Module R W] [TopologicalSpace Q] [AddTorsor W Q] [TopologicalSpace W] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (f : P →ᴬ[R] W) (g : P →ᴬ[R] Q) : ↑(f +ᵥ g) = ↑f +ᵥ ↑g
ContinuousAffineMap.linear_decompAffineEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
(R : Type u_1) (S : Type u_2) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring S] [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [Module S W] [SMulCommClass R S W] [ContinuousConstSMul S W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] : (ContinuousAffineMap.decompAffineEquiv R S V Q).linear = ContinuousAffineMap.decompLinearEquiv R S V W
ContinuousAffineMap.snd_decompAffineEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
(R : Type u_1) (S : Type u_2) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring S] [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [Module S W] [SMulCommClass R S W] [ContinuousConstSMul S W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (f : V →ᴬ[R] Q) : ((ContinuousAffineMap.decompAffineEquiv R S V Q) f).2 = f.contLinear
ContinuousAffineMap.fst_decompAffineEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
(R : Type u_1) (S : Type u_2) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring S] [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [Module S W] [SMulCommClass R S W] [ContinuousConstSMul S W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (f : V →ᴬ[R] Q) : ((ContinuousAffineMap.decompAffineEquiv R S V Q) f).1 = f 0
ContinuousAffineMap.decompAffineEquiv_symm_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
(R : Type u_1) (S : Type u_2) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring S] [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [Module S W] [SMulCommClass R S W] [ContinuousConstSMul S W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (p : Q × (V →L[R] W)) : ((ContinuousAffineMap.decompAffineEquiv R S V Q).symm p).contLinear = p.2
ContinuousAffineMap.decompAffineEquiv_symm_apply 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
(R : Type u_1) (S : Type u_2) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring S] [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [Module S W] [SMulCommClass R S W] [ContinuousConstSMul S W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (p : Q × (V →L[R] W)) (x : V) : ((ContinuousAffineMap.decompAffineEquiv R S V Q).symm p) x = p.2 x +ᵥ p.1
ContinuousAffineEquiv.pointReflection 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (x : P₁) : P₁ ≃ᴬ[k] P₁
ContinuousAffineEquiv.constVSub 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (p : P₁) : P₁ ≃ᴬ[k] V₁
ContinuousAffineEquiv.vaddConst 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (p : P₁) : V₁ ≃ᴬ[k] P₁
ContinuousAffineEquiv.pointReflection_involutive 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (x : P₁) : Function.Involutive ⇑(ContinuousAffineEquiv.pointReflection k x)
ContinuousAffineEquiv.pointReflection_self 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (x : P₁) : (ContinuousAffineEquiv.pointReflection k x) x = x
ContinuousAffineEquiv.toAffineEquiv_pointReflection 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (x : P₁) : ↑(ContinuousAffineEquiv.pointReflection k x) = AffineEquiv.pointReflection k x
ContinuousAffineEquiv.pointReflection_symm 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (x : P₁) : (ContinuousAffineEquiv.pointReflection k x).symm = ContinuousAffineEquiv.pointReflection k x
ContinuousAffineEquiv.toAffineEquiv_constVSub 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] {p : P₁} : ↑(ContinuousAffineEquiv.constVSub k p) = AffineEquiv.constVSub k p
ContinuousAffineEquiv.toAffineEquiv_vaddConst 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] {p : P₁} : ↑(ContinuousAffineEquiv.vaddConst k p) = AffineEquiv.vaddConst k p
ContinuousAffineEquiv.constVSub_apply 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (p x✝ : P₁) : (ContinuousAffineEquiv.constVSub k p) x✝ = p -ᵥ x✝
ContinuousAffineEquiv.coe_pointReflection 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (x : P₁) : ⇑(ContinuousAffineEquiv.pointReflection k x) = ⇑(Equiv.pointReflection x)
ContinuousAffineEquiv.vaddConst_symm_apply 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (p p' : P₁) : (ContinuousAffineEquiv.vaddConst k p).symm p' = p' -ᵥ p
ContinuousAffineEquiv.vaddConst_apply 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (p : P₁) (v : V₁) : (ContinuousAffineEquiv.vaddConst k p) v = v +ᵥ p
ContinuousAffineEquiv.pointReflection_apply 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (x y : P₁) : (ContinuousAffineEquiv.pointReflection k x) y = (x -ᵥ y) +ᵥ x
ContinuousAffineEquiv.constVSub_symm_apply 📋 Mathlib.Topology.Algebra.ContinuousAffineEquiv
(k : Type u_1) {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [TopologicalSpace V₁] [IsTopologicalAddTorsor P₁] (p : P₁) (x✝ : V₁) : (ContinuousAffineEquiv.constVSub k p).symm x✝ = -x✝ +ᵥ p
AffineSubspace.isClosed_direction_iff 📋 Mathlib.Topology.Algebra.AffineSubspace
{R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [TopologicalSpace V] [IsTopologicalAddTorsor P] [T1Space V] (s : AffineSubspace R P) : IsClosed ↑s.direction ↔ IsClosed ↑s
AffineSubspace.instIsTopologicalAddTorsorSubtypeMemSubmoduleDirection 📋 Mathlib.Topology.Algebra.AffineSubspace
{R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P] [AddTorsor V P] [TopologicalSpace V] [IsTopologicalAddTorsor P] {s : AffineSubspace R P} [Nonempty ↥s] : IsTopologicalAddTorsor ↥s
asymptoticCone_closure 📋 Mathlib.Topology.Algebra.AsymptoticCone
{k : Type u_1} {V : Type u_2} {P : Type u_3} [Field k] [LinearOrder k] [AddCommGroup V] [Module k V] [AddTorsor V P] [TopologicalSpace V] [TopologicalSpace k] [OrderTopology k] [IsStrictOrderedRing k] [IsTopologicalAddGroup V] [ContinuousSMul k V] [TopologicalSpace P] [IsTopologicalAddTorsor P] (s : Set P) : asymptoticCone k (closure s) = asymptoticCone k s
AffineSpace.asymptoticNhds_bind_nhds 📋 Mathlib.Topology.Algebra.AsymptoticCone
{k : Type u_1} {V : Type u_2} {P : Type u_3} [Field k] [LinearOrder k] [AddCommGroup V] [Module k V] [AddTorsor V P] [TopologicalSpace V] [TopologicalSpace k] [OrderTopology k] [IsStrictOrderedRing k] [IsTopologicalAddGroup V] [ContinuousSMul k V] [TopologicalSpace P] [IsTopologicalAddTorsor P] (v : V) : (AffineSpace.asymptoticNhds k P v).bind nhds = AffineSpace.asymptoticNhds k P v
ContinuousAffineMap.instTopologicalSpace 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor P] [IsTopologicalAddTorsor Q] : TopologicalSpace (P →ᴬ[R] Q)
ContinuousAffineMap.instRegularSpace 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor P] [IsTopologicalAddTorsor Q] : RegularSpace (P →ᴬ[R] Q)
ContinuousAffineMap.instT0Space 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor P] [IsTopologicalAddTorsor Q] [T0Space W] : T0Space (P →ᴬ[R] Q)
ContinuousAffineMap.continuous_const 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor P] [IsTopologicalAddTorsor Q] : Continuous (ContinuousAffineMap.const R P)
ContinuousAffineMap.instContinuousEvalConst 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor P] [IsTopologicalAddTorsor Q] : ContinuousEvalConst (P →ᴬ[R] Q) P Q
ContinuousAffineMap.instIsTopologicalAddGroup 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [IsTopologicalAddTorsor P] [IsTopologicalAddGroup W] : IsTopologicalAddGroup (P →ᴬ[R] W)
ContinuousAffineMap.continuous_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor P] [IsTopologicalAddTorsor Q] [IsTopologicalAddGroup W] : Continuous ContinuousAffineMap.contLinear
ContinuousAffineMap.instIsTopologicalAddTorsor 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor P] [IsTopologicalAddTorsor Q] [IsTopologicalAddGroup W] : IsTopologicalAddTorsor (P →ᴬ[R] Q)
ContinuousAffineMap.continuous_rng 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor P] [IsTopologicalAddTorsor Q] {α : Type u_6} [TopologicalSpace α] [IsTopologicalAddGroup W] {f : α → P →ᴬ[R] Q} (h₁ : ∀ (p : P), Continuous fun x => (f x) p) (h₂ : Continuous fun x => (f x).contLinear) : Continuous f
ContinuousAffineMap.instContinuousConstSMul 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [IsTopologicalAddTorsor P] [IsTopologicalAddGroup W] {S : Type u_6} [Monoid S] [DistribMulAction S W] [SMulCommClass R S W] [ContinuousConstSMul S W] : ContinuousConstSMul S (P →ᴬ[R] W)
ContinuousAffineMap.continuous_rng_iff 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor P] [IsTopologicalAddTorsor Q] {α : Type u_6} [TopologicalSpace α] [IsTopologicalAddGroup W] (f : α → P →ᴬ[R] Q) : Continuous f ↔ (∀ (p : P), Continuous fun x => (f x) p) ∧ Continuous fun x => (f x).contLinear
ContinuousAffineMap.decompHomeomorph 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
(R : Type u_1) (V : Type u_2) {W : Type u_3} (Q : Type u_5) [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup V] [ContinuousSMul R V] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] : (V →ᴬ[R] Q) ≃ₜ Q × (V →L[R] W)
ContinuousAffineMap.instContinuousSMul 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [IsTopologicalAddTorsor P] [IsTopologicalAddGroup W] [ContinuousSMul R W] : ContinuousSMul R (P →ᴬ[R] W)
ContinuousAffineMap.continuous_rng_of_exists 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
{R : Type u_1} {V : Type u_2} {W : Type u_3} {P : Type u_4} {Q : Type u_5} [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor P] [IsTopologicalAddTorsor Q] {α : Type u_6} [TopologicalSpace α] [IsTopologicalAddGroup W] [ContinuousSMul R V] {f : α → P →ᴬ[R] Q} (h₁ : ∃ p, Continuous fun x => (f x) p) (h₂ : Continuous fun x => (f x).contLinear) : Continuous f
ContinuousAffineMap.fst_decompHomeomorph 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
(R : Type u_1) (V : Type u_2) {W : Type u_3} (Q : Type u_5) [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup V] [ContinuousSMul R V] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (f : V →ᴬ[R] Q) : ((ContinuousAffineMap.decompHomeomorph R V Q) f).1 = f 0
ContinuousAffineMap.snd_decompHomeomorph 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
(R : Type u_1) (V : Type u_2) {W : Type u_3} (Q : Type u_5) [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup V] [ContinuousSMul R V] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (f : V →ᴬ[R] Q) : ((ContinuousAffineMap.decompHomeomorph R V Q) f).2 = f.contLinear
ContinuousAffineMap.decompContinuousAffineEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
(R : Type u_1) (V : Type u_2) {W : Type u_3} (Q : Type u_5) [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup V] [ContinuousSMul R V] [IsTopologicalAddGroup W] [ContinuousConstSMul R W] [IsTopologicalAddTorsor Q] : (V →ᴬ[R] Q) ≃ᴬ[R] Q × (V →L[R] W)
ContinuousAffineMap.decompHomeomorph_symm_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
(R : Type u_1) (V : Type u_2) {W : Type u_3} (Q : Type u_5) [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup V] [ContinuousSMul R V] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (p : Q × (V →L[R] W)) : ((ContinuousAffineMap.decompHomeomorph R V Q).symm p).contLinear = p.2
ContinuousAffineMap.decompHomeomorph_symm_apply 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
(R : Type u_1) (V : Type u_2) {W : Type u_3} (Q : Type u_5) [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup V] [ContinuousSMul R V] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (p : Q × (V →L[R] W)) (x : V) : ((ContinuousAffineMap.decompHomeomorph R V Q).symm p) x = p.2 x +ᵥ p.1
ContinuousAffineMap.fst_decompContinuousAffineEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
(R : Type u_1) (V : Type u_2) {W : Type u_3} (Q : Type u_5) [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup V] [ContinuousSMul R V] [IsTopologicalAddGroup W] [ContinuousConstSMul R W] [IsTopologicalAddTorsor Q] (f : V →ᴬ[R] Q) : ((ContinuousAffineMap.decompContinuousAffineEquiv R V Q) f).1 = f 0
ContinuousAffineMap.snd_decompContinuousAffineEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
(R : Type u_1) (V : Type u_2) {W : Type u_3} (Q : Type u_5) [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup V] [ContinuousSMul R V] [IsTopologicalAddGroup W] [ContinuousConstSMul R W] [IsTopologicalAddTorsor Q] (f : V →ᴬ[R] Q) : ((ContinuousAffineMap.decompContinuousAffineEquiv R V Q) f).2 = f.contLinear
ContinuousAffineMap.decompContinuousAffineEquiv_symm_contLinear 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
(R : Type u_1) (V : Type u_2) {W : Type u_3} (Q : Type u_5) [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup V] [ContinuousSMul R V] [IsTopologicalAddGroup W] [ContinuousConstSMul R W] [IsTopologicalAddTorsor Q] (p : Q × (V →L[R] W)) : ((ContinuousAffineMap.decompContinuousAffineEquiv R V Q).symm p).contLinear = p.2
ContinuousAffineMap.decompContinuousAffineEquiv_symm_apply 📋 Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
(R : Type u_1) (V : Type u_2) {W : Type u_3} (Q : Type u_5) [NormedField R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup V] [ContinuousSMul R V] [IsTopologicalAddGroup W] [ContinuousConstSMul R W] [IsTopologicalAddTorsor Q] (p : Q × (V →L[R] W)) (x : V) : ((ContinuousAffineMap.decompContinuousAffineEquiv R V Q).symm p) x = p.2 x +ᵥ p.1
instProperVAdd 📋 Mathlib.Topology.Algebra.ProperAction.Torsor
{V : Type u_1} {P : Type u_2} [AddGroup V] [AddTorsor V P] [TopologicalSpace V] [TopologicalSpace P] [IsTopologicalAddTorsor P] : ProperVAdd V P

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