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Result

Found 238 declarations mentioning Isometry. Of these, only the first 200 are shown.

• Isometry 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α → β) : Prop
• isometry_id 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} [PseudoEMetricSpace α] : Isometry id
• NNReal.isometry_coe 📋 Mathlib.Topology.MetricSpace.Isometry
  : Isometry NNReal.toReal
• isometry_subsingleton 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} [Subsingleton α] : Isometry f
• PseudoEMetricSpace.isometry_induced 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} (f : α → β) [m : PseudoEMetricSpace β] : Isometry f
• Isometry.injective 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) : Function.Injective f
• PseudoMetricSpace.isometry_induced 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} (f : α → β) [m : PseudoMetricSpace β] : Isometry f
• IsometryEquiv.mk 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (toEquiv : α ≃ β) (isometry_toFun : Isometry toEquiv.toFun) : α ≃ᵢ β
• Isometry.isUniformInducing 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) : IsUniformInducing f
• Isometry.uniformContinuous 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) : UniformContinuous f
• EMetricSpace.isometry_induced 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} (f : α → β) (hf : Function.Injective f) [m : EMetricSpace β] : Isometry f
• IsometryEquiv.isometry_toFun 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (self : α ≃ᵢ β) : Isometry self.toFun
• Isometry.antilipschitz 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) : AntilipschitzWith 1 f
• Isometry.antilipschitzWith 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) : AntilipschitzWith 1 f
• Isometry.lipschitz 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) : LipschitzWith 1 f
• Isometry.lipschitzWith 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) : LipschitzWith 1 f
• Isometry.right_inv 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} {g : β → α} (h : Isometry f) (hg : Function.RightInverse g f) : Isometry g
• Isometry.continuous 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) : Continuous f
• Isometry.isUniformEmbedding 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) : IsUniformEmbedding f
• MetricSpace.isometry_induced 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} (f : α → β) (hf : Function.Injective f) [m : MetricSpace β] : Isometry f
• Isometry.diam_range 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) : Metric.diam (Set.range f) = Metric.diam Set.univ
• Isometry.isEmbedding 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) : Topology.IsEmbedding f
• IsometryEquiv.mk' 📋 Mathlib.Topology.MetricSpace.Isometry
  {β : Type v} [PseudoEMetricSpace β] {α : Type u} [EMetricSpace α] (f : α → β) (g : β → α) (hfg : ∀ (x : β), f (g x) = x) (hf : Isometry f) : α ≃ᵢ β
• IsometryClass.isometry 📋 Mathlib.Topology.MetricSpace.Isometry
  {F : Type u_3} {α : outParam (Type u_4)} {β : outParam (Type u_5)} {inst✝ : PseudoEMetricSpace α} {inst✝¹ : PseudoEMetricSpace β} {inst✝² : FunLike F α β} [self : IsometryClass F α β] (f : F) : Isometry ⇑f
• IsometryClass.mk 📋 Mathlib.Topology.MetricSpace.Isometry
  {F : Type u_3} {α : outParam (Type u_4)} {β : outParam (Type u_5)} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [FunLike F α β] (isometry : ∀ (f : F), Isometry ⇑f) : IsometryClass F α β
• Isometry.diam_image 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (s : Set α) : Metric.diam (f '' s) = Metric.diam s
• Isometry.mapsTo_ball 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ) : Set.MapsTo f (Metric.ball x r) (Metric.ball (f x) r)
• Isometry.mapsTo_closedBall 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ) : Set.MapsTo f (Metric.closedBall x r) (Metric.closedBall (f x) r)
• Isometry.mapsTo_sphere 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ) : Set.MapsTo f (Metric.sphere x r) (Metric.sphere (f x) r)
• Isometry.single 📋 Mathlib.Topology.MetricSpace.Isometry
  {ι : Type u_2} [Fintype ι] [DecidableEq ι] {E : ι → Type u_3} [(i : ι) → PseudoEMetricSpace (E i)] [(i : ι) → Zero (E i)] (i : ι) : Isometry (Pi.single i)
• Isometry.comp 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {g : β → γ} {f : α → β} (hg : Isometry g) (hf : Isometry f) : Isometry (g ∘ f)
• IsUniformEmbedding.to_isometry 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u_3} {β : Type u_4} [UniformSpace α] [MetricSpace β] {f : α → β} (h : IsUniformEmbedding f) : Isometry f
• Isometry.dist_eq 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : Isometry f → ∀ (x y : α), dist (f x) (f y) = dist x y
• Isometry.nndist_eq 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : Isometry f → ∀ (x y : α), nndist (f x) (f y) = nndist x y
• Isometry.of_dist_eq 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : (∀ (x y : α), dist (f x) (f y) = dist x y) → Isometry f
• Isometry.of_nndist_eq 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : (∀ (x y : α), nndist (f x) (f y) = nndist x y) → Isometry f
• Isometry.preimage_ball 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ) : f ⁻¹' Metric.ball (f x) r = Metric.ball x r
• Isometry.preimage_closedBall 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ) : f ⁻¹' Metric.closedBall (f x) r = Metric.closedBall x r
• Isometry.preimage_sphere 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ℝ) : f ⁻¹' Metric.sphere (f x) r = Metric.sphere x r
• isometry_iff_dist_eq 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : Isometry f ↔ ∀ (x y : α), dist (f x) (f y) = dist x y
• isometry_iff_nndist_eq 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : Isometry f ↔ ∀ (x y : α), nndist (f x) (f y) = nndist x y
• IsometryEquiv.isometry 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) : Isometry ⇑h
• Isometry.lipschitzWith_iff 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u_3} {β : Type u_4} {γ : Type u_5} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {f : α → β} {g : β → γ} (K : NNReal) (h : Isometry g) : LipschitzWith K (g ∘ f) ↔ LipschitzWith K f
• isometry_subtype_coe 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} [PseudoEMetricSpace α] {s : Set α} : Isometry Subtype.val
• Isometry.isometryEquivOnRange 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) : α ≃ᵢ ↑(Set.range f)
• Isometry.isClosedEmbedding 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {γ : Type w} [EMetricSpace α] [CompleteSpace α] [EMetricSpace γ] {f : α → γ} (hf : Isometry f) : Topology.IsClosedEmbedding f
• Isometry.postcomp_pi 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace β] [PseudoEMetricSpace γ] [Fintype α] {g : β → γ} (hg : Isometry g) : Isometry fun f => g ∘ f
• Topology.IsEmbedding.to_isometry 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u_3} {β : Type u_4} [TopologicalSpace α] [PseudoMetricSpace β] {f : α → β} (h : Topology.IsEmbedding f) : Isometry f
• Isometry.comp_continuous_iff 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} {γ : Type u_3} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} : Continuous (f ∘ g) ↔ Continuous g
• Isometry.ediam_range 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) : Metric.ediam (Set.range f) = Metric.ediam Set.univ
• Isometry.ediam_image 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) (s : Set α) : Metric.ediam (f '' s) = Metric.ediam s
• Isometry.edist_eq 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) (x y : α) : edist (f x) (f y) = edist x y
• Isometry.locallyLipschitzOn_image 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u_3} {β : Type u_4} {γ : Type u_5} [EMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {g : α → β} {h : β → γ} {s : Set α} (hg : Isometry g) (hL : LocallyLipschitzOn s (h ∘ g)) : LocallyLipschitzOn (g '' s) h
• Isometry.comp_continuousOn_iff 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} {γ : Type u_3} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} {s : Set γ} : ContinuousOn (f ∘ g) s ↔ ContinuousOn g s
• Isometry.mapsTo_closedEBall 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ENNReal) : Set.MapsTo f (Metric.closedEBall x r) (Metric.closedEBall (f x) r)
• Isometry.mapsTo_eball 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (r : ENNReal) : Set.MapsTo f (Metric.eball x r) (Metric.eball (f x) r)
• Isometry.preimage_setOfPred_dist 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (p : ℝ → Prop) : f ⁻¹' {y | p (dist y (f x))} = {y | p (dist y x)}
• Isometry.preimage_setOf_dist 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (hf : Isometry f) (x : α) (p : ℝ → Prop) : f ⁻¹' {y | p (dist y (f x))} = {y | p (dist y x)}
• Isometry.preimage_closedEBall 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) (x : α) (r : ENNReal) : f ⁻¹' Metric.closedEBall (f x) r = Metric.closedEBall x r
• Isometry.preimage_eball 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) (x : α) (r : ENNReal) : f ⁻¹' Metric.eball (f x) r = Metric.eball x r
• Isometry.prodMap 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {δ : Type u_3} [PseudoEMetricSpace δ] {f : α → β} {g : γ → δ} (hf : Isometry f) (hg : Isometry g) : Isometry (Prod.map f g)
• Isometry.tendsto_nhds_iff 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {ι : Type u_3} {f : α → β} {g : ι → α} {a : Filter ι} {b : α} (hf : Isometry f) : Filter.Tendsto g a (nhds b) ↔ Filter.Tendsto (f ∘ g) a (nhds (f b))
• Isometry.piMap 📋 Mathlib.Topology.MetricSpace.Isometry
  {ι : Type u_5} [Fintype ι] {α : ι → Type u_3} {β : ι → Type u_4} [(i : ι) → PseudoEMetricSpace (α i)] [(i : ι) → PseudoEMetricSpace (β i)] (f : (i : ι) → α i → β i) (hf : ∀ (i : ι), Isometry (f i)) : Isometry (Pi.map f)
• Isometry.inl 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [AddZeroClass α] [AddZeroClass β] : Isometry ⇑(AddMonoidHom.inl α β)
• Isometry.inr 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [AddZeroClass α] [AddZeroClass β] : Isometry ⇑(AddMonoidHom.inr α β)
• IsometryEquiv.coe_mk 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (e : α ≃ β) (h : Isometry e.toFun) : ⇑{ toEquiv := e, isometry_toFun := h } = ⇑e
• Isometry.isometryEquivOnRange_toEquiv 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) : h.isometryEquivOnRange.toEquiv = Equiv.ofInjective f ⋯
• Isometry.isometryEquivOnRange_apply 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β} (h : Isometry f) (a : α) : h.isometryEquivOnRange a = ⟨f a, ⋯⟩
• isometry_add_left 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} [Add M] [PseudoEMetricSpace M] [IsIsometricVAdd M M] (a : M) : Isometry fun x => a + x
• isometry_mul_left 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} [Mul M] [PseudoEMetricSpace M] [IsIsometricSMul M M] (a : M) : Isometry fun x => a * x
• isometry_add_right 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} [Add M] [PseudoEMetricSpace M] [IsIsometricVAdd Mᵃᵒᵖ M] (a : M) : Isometry fun x => x + a
• isometry_mul_right 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} [Mul M] [PseudoEMetricSpace M] [IsIsometricSMul Mᵐᵒᵖ M] (a : M) : Isometry fun x => x * a
• IsIsometricSMul.isometry_smul 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} (X : Type w) {inst✝ : PseudoEMetricSpace X} {inst✝¹ : SMul M X} [self : IsIsometricSMul M X] (c : M) : Isometry fun x => c • x
• IsIsometricSMul.mk 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} {X : Type w} [PseudoEMetricSpace X] [SMul M X] (isometry_smul : ∀ (c : M), Isometry fun x => c • x) : IsIsometricSMul M X
• IsIsometricVAdd.isometry_vadd 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} (X : Type w) {inst✝ : PseudoEMetricSpace X} {inst✝¹ : VAdd M X} [self : IsIsometricVAdd M X] (c : M) : Isometry fun x => c +ᵥ x
• IsIsometricVAdd.mk 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} {X : Type w} [PseudoEMetricSpace X] [VAdd M X] (isometry_vadd : ∀ (c : M), Isometry fun x => c +ᵥ x) : IsIsometricVAdd M X
• isometry_inv 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {G : Type v} [Group G] [PseudoEMetricSpace G] [IsIsometricSMul G G] [IsIsometricSMul Gᵐᵒᵖ G] : Isometry Inv.inv
• isometry_neg 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {G : Type v} [AddGroup G] [PseudoEMetricSpace G] [IsIsometricVAdd G G] [IsIsometricVAdd Gᵃᵒᵖ G] : Isometry Neg.neg
• Real.isometry_intCast 📋 Mathlib.Analysis.Normed.Group.Uniform
  : Isometry Int.cast
• Isometry.nnnorm_map_of_map_one 📋 Mathlib.Analysis.Normed.Group.Uniform
  {E : Type u_2} {F : Type u_3} [SeminormedGroup E] [SeminormedGroup F] {f : E → F} (hi : Isometry f) (h₁ : f 1 = 1) (x : E) : ‖f x‖₊ = ‖x‖₊
• Isometry.nnnorm_map_of_map_zero 📋 Mathlib.Analysis.Normed.Group.Uniform
  {E : Type u_2} {F : Type u_3} [SeminormedAddGroup E] [SeminormedAddGroup F] {f : E → F} (hi : Isometry f) (h₁ : f 0 = 0) (x : E) : ‖f x‖₊ = ‖x‖₊
• Isometry.norm_map_of_map_one 📋 Mathlib.Analysis.Normed.Group.Uniform
  {E : Type u_2} {F : Type u_3} [SeminormedGroup E] [SeminormedGroup F] {f : E → F} (hi : Isometry f) (h₁ : f 1 = 1) (x : E) : ‖f x‖ = ‖x‖
• Isometry.norm_map_of_map_zero 📋 Mathlib.Analysis.Normed.Group.Uniform
  {E : Type u_2} {F : Type u_3} [SeminormedAddGroup E] [SeminormedAddGroup F] {f : E → F} (hi : Isometry f) (h₁ : f 0 = 0) (x : E) : ‖f x‖ = ‖x‖
• AddMonoidHomClass.isometry_of_norm 📋 Mathlib.Analysis.Normed.Group.Uniform
  {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [SeminormedAddGroup E] [SeminormedAddGroup F] [FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] (f : 𝓕) : (∀ (x : E), ‖f x‖ = ‖x‖) → Isometry ⇑f
• MonoidHomClass.isometry_of_norm 📋 Mathlib.Analysis.Normed.Group.Uniform
  {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [SeminormedGroup E] [SeminormedGroup F] [FunLike 𝓕 E F] [MonoidHomClass 𝓕 E F] (f : 𝓕) : (∀ (x : E), ‖f x‖ = ‖x‖) → Isometry ⇑f
• AddMonoidHomClass.isometry_iff_norm 📋 Mathlib.Analysis.Normed.Group.Uniform
  {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [SeminormedAddGroup E] [SeminormedAddGroup F] [FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] (f : 𝓕) : Isometry ⇑f ↔ ∀ (x : E), ‖f x‖ = ‖x‖
• MonoidHomClass.isometry_iff_norm 📋 Mathlib.Analysis.Normed.Group.Uniform
  {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [SeminormedGroup E] [SeminormedGroup F] [FunLike 𝓕 E F] [MonoidHomClass 𝓕 E F] (f : 𝓕) : Isometry ⇑f ↔ ∀ (x : E), ‖f x‖ = ‖x‖
• Isometry.toDilation 📋 Mathlib.Topology.MetricSpace.Dilation
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α → β) (hf : Isometry f) : α →ᵈ β
• Isometry.toDilation_toFun 📋 Mathlib.Topology.MetricSpace.Dilation
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α → β) (hf : Isometry f) (a✝ : α) : (Isometry.toDilation f hf) a✝ = f a✝
• Isometry.toDilation_ratio 📋 Mathlib.Topology.MetricSpace.Dilation
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} {hf : Isometry f} : Dilation.ratio (Isometry.toDilation f hf) = 1
• RingHom.isometry 📋 Mathlib.Analysis.Normed.Ring.Lemmas
  {𝕜₁ : Type u_3} {𝕜₂ : Type u_4} [SeminormedRing 𝕜₁] [SeminormedRing 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) [RingHomIsometric σ] : Isometry ⇑σ
• algebraMap_isometry 📋 Mathlib.Analysis.Normed.Module.Basic
  (𝕜 : Type u_1) (𝕜' : Type u_2) [NormedField 𝕜] [SeminormedRing 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormOneClass 𝕜'] : Isometry ⇑(algebraMap 𝕜 𝕜')
• Metric.infEDist_image 📋 Mathlib.Topology.MetricSpace.HausdorffDistance
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {x : α} {t : Set α} {Φ : α → β} (hΦ : Isometry Φ) : Metric.infEDist (Φ x) (Φ '' t) = Metric.infEDist x t
• Metric.hausdorffEDist_image 📋 Mathlib.Topology.MetricSpace.HausdorffDistance
  {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {s t : Set α} {Φ : α → β} (h : Isometry Φ) : Metric.hausdorffEDist (Φ '' s) (Φ '' t) = Metric.hausdorffEDist s t
• Metric.infDist_image 📋 Mathlib.Topology.MetricSpace.HausdorffDistance
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {t : Set α} {x : α} {Φ : α → β} (hΦ : Isometry Φ) : Metric.infDist (Φ x) (Φ '' t) = Metric.infDist x t
• Metric.hausdorffDist_image 📋 Mathlib.Topology.MetricSpace.HausdorffDistance
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {s t : Set α} {Φ : α → β} (h : Isometry Φ) : Metric.hausdorffDist (Φ '' s) (Φ '' t) = Metric.hausdorffDist s t
• Metric.PiNatEmbed.isometry_embed 📋 Mathlib.Topology.MetricSpace.PiNat
  {ι : Type u_2} {X : Type u_3} {Y : ι → Type u_4} {f : (i : ι) → X → Y i} [Encodable ι] [(i : ι) → PseudoEMetricSpace (Y i)] : Isometry (Metric.PiNatEmbed.embed X Y f)
• Metric.isometry_inl 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] : Isometry Sum.inl
• Metric.isometry_inr 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] : Isometry Sum.inr
• Metric.Sigma.isometry_mk 📋 Mathlib.Topology.MetricSpace.Gluing
  {ι : Type u_1} {E : ι → Type u_2} [(i : ι) → MetricSpace (E i)] (i : ι) : Isometry (Sigma.mk i)
• Metric.GlueSpace 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : Type (max u v)
• Metric.gluePremetric 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : PseudoMetricSpace (X ⊕ Y)
• Metric.instMetricSpaceGlueSpace 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : MetricSpace (Metric.GlueSpace hΦ hΨ)
• Metric.toGlueL 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ) (x : X) : Metric.GlueSpace hΦ hΨ
• Metric.toGlueR 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ) (y : Y) : Metric.GlueSpace hΦ hΨ
• Metric.inhabitedLeft 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ) [Inhabited X] : Inhabited (Metric.GlueSpace hΦ hΨ)
• Metric.inhabitedRight 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ) [Inhabited Y] : Inhabited (Metric.GlueSpace hΦ hΨ)
• Metric.InductiveLimit 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} (I : ∀ (n : ℕ), Isometry (f n)) : Type u
• Metric.inductivePremetric 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} (I : ∀ (n : ℕ), Isometry (f n)) : PseudoMetricSpace ((n : ℕ) × X n)
• Metric.instMetricSpaceInductiveLimit 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} {I : ∀ (n : ℕ), Isometry (f n)} : MetricSpace (Metric.InductiveLimit I)
• Metric.toInductiveLimit 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} (I : ∀ (n : ℕ), Isometry (f n)) (n : ℕ) (x : X n) : Metric.InductiveLimit I
• Metric.instInhabitedInductiveLimitOfOfNatNat 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} (I : ∀ (n : ℕ), Isometry (f n)) [Inhabited (X 0)] : Inhabited (Metric.InductiveLimit I)
• Metric.toGlueL_isometry 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : Isometry (Metric.toGlueL hΦ hΨ)
• Metric.toGlueR_isometry 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : Isometry (Metric.toGlueR hΦ hΨ)
• Metric.toGlue_commute 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : Metric.toGlueL hΦ hΨ ∘ Φ = Metric.toGlueR hΦ hΨ ∘ Ψ
• Metric.toInductiveLimit_commute 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} (I : ∀ (n : ℕ), Isometry (f n)) (n : ℕ) : Metric.toInductiveLimit I n.succ ∘ f n = Metric.toInductiveLimit I n
• Metric.toInductiveLimit_isometry 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} (I : ∀ (n : ℕ), Isometry (f n)) (n : ℕ) : Isometry (Metric.toInductiveLimit I n)
• Metric.separableSpaceInductiveLimit_of_separableSpace 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] [hs : ∀ (n : ℕ), TopologicalSpace.SeparableSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} (I : ∀ (n : ℕ), Isometry (f n)) : TopologicalSpace.SeparableSpace (Metric.InductiveLimit I)
• Metric.dense_iUnion_range_toInductiveLimit 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} (I : ∀ (n : ℕ), Isometry (f n)) : Dense (⋃ i, Set.range (Metric.toInductiveLimit I i))
• Metric.inductiveLimitDist_eq_dist 📋 Mathlib.Topology.MetricSpace.Gluing
  {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} (I : ∀ (n : ℕ), Isometry (f n)) (x y : (n : ℕ) × X n) (m : ℕ) (hx : x.fst ≤ m) (hy : y.fst ≤ m) : Metric.inductiveLimitDist f x y = dist (Nat.leRecOn hx (fun {k} => f k) x.snd) (Nat.leRecOn hy (fun {k} => f k) y.snd)
• NormedAddGroupHom.isometry_id 📋 Mathlib.Analysis.Normed.Group.Hom
  {V : Type u_1} [SeminormedAddCommGroup V] : Isometry ⇑(NormedAddGroupHom.id V)
• NormedAddGroupHom.normNoninc_of_isometry 📋 Mathlib.Analysis.Normed.Group.Hom
  {V : Type u_1} {W : Type u_2} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] {f : NormedAddGroupHom V W} (hf : Isometry ⇑f) : f.NormNoninc
• NormedAddGroupHom.norm_eq_of_isometry 📋 Mathlib.Analysis.Normed.Group.Hom
  {V : Type u_1} {W : Type u_2} [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] {f : NormedAddGroupHom V W} (hf : Isometry ⇑f) (v : V) : ‖f v‖ = ‖v‖
• NormedAddGroupHom.isometry_comp 📋 Mathlib.Analysis.Normed.Group.Hom
  {V₁ : Type u_3} {V₂ : Type u_4} {V₃ : Type u_5} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] {g : NormedAddGroupHom V₂ V₃} {f : NormedAddGroupHom V₁ V₂} (hg : Isometry ⇑g) (hf : Isometry ⇑f) : Isometry ⇑(g.comp f)
• SemilinearIsometryClass.isometry 📋 Mathlib.Analysis.Normed.Operator.LinearIsometry
  {R : Type u_1} {R₂ : Type u_2} {E : Type u_4} {E₂ : Type u_5} {𝓕 : Type u_8} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] [FunLike 𝓕 E E₂] [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) : Isometry ⇑f
• LinearIsometry.isometry 📋 Mathlib.Analysis.Normed.Operator.LinearIsometry
  {R : Type u_1} {R₂ : Type u_2} {E : Type u_4} {E₂ : Type u_5} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) : Isometry ⇑f
• LinearMap.toLinearIsometry 📋 Mathlib.Analysis.Normed.Operator.LinearIsometry
  {R : Type u_1} {R₂ : Type u_2} {E : Type u_4} {E₂ : Type u_5} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (f : E →ₛₗ[σ₁₂] E₂) (hf : Isometry ⇑f) : E →ₛₗᵢ[σ₁₂] E₂
• LinearIsometryEquiv.isometry 📋 Mathlib.Analysis.Normed.Operator.LinearIsometry
  {R : Type u_1} {R₂ : Type u_2} {E : Type u_4} {E₂ : Type u_5} [Semiring R] [Semiring R₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup E₂] [Module R E] [Module R₂ E₂] (e : E ≃ₛₗᵢ[σ₁₂] E₂) : Isometry ⇑e
• MulOpposite.isometry_opLinearEquiv 📋 Mathlib.Analysis.Normed.Operator.LinearIsometry
  {R : Type u_9} {H : Type u_10} [Semiring R] [SeminormedAddCommGroup H] [Module R H] : Isometry ⇑(MulOpposite.opLinearEquiv R)
• star_isometry 📋 Mathlib.Analysis.CStarAlgebra.Basic
  {E : Type u_2} [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] : Isometry star
• Complex.isometry_ofReal 📋 Mathlib.Analysis.Complex.Basic
  : Isometry Complex.ofReal
• Complex.isometry_intCast 📋 Mathlib.Analysis.Complex.Basic
  : Isometry Int.cast
• Complex.isometry_conj 📋 Mathlib.Analysis.Complex.Basic
  : Isometry ⇑(starRingEnd ℂ)
• MeasureTheory.Lp.isometry_compMeasurePreserving 📋 Mathlib.MeasureTheory.Function.LpSpace.Basic
  {α : Type u_1} {E : Type u_4} {m : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : MeasureTheory.Measure β} {f : α → β} [Fact (1 ≤ p)] (hf : MeasureTheory.MeasurePreserving f μ μb) : Isometry ⇑(MeasureTheory.Lp.compMeasurePreserving f hf)
• ContinuousLinearMap.isometry_mul 📋 Mathlib.Analysis.Normed.Operator.Mul
  (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (R : Type u_3) [NonUnitalSeminormedRing R] [NormedSpace 𝕜 R] [IsScalarTower 𝕜 R R] [SMulCommClass 𝕜 R R] [RegularNormedAlgebra 𝕜 R] : Isometry ⇑(ContinuousLinearMap.mul 𝕜 R)
• RegularNormedAlgebra.isometry_mul' 📋 Mathlib.Analysis.Normed.Operator.Mul
  {𝕜 : Type u_1} {inst✝ : NontriviallyNormedField 𝕜} {R : Type u_3} {inst✝¹ : NonUnitalSeminormedRing R} {inst✝² : NormedSpace 𝕜 R} {inst✝³ : IsScalarTower 𝕜 R R} {inst✝⁴ : SMulCommClass 𝕜 R R} [self : RegularNormedAlgebra 𝕜 R] : Isometry ⇑(ContinuousLinearMap.mul 𝕜 R)
• RegularNormedAlgebra.mk 📋 Mathlib.Analysis.Normed.Operator.Mul
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {R : Type u_3} [NonUnitalSeminormedRing R] [NormedSpace 𝕜 R] [IsScalarTower 𝕜 R R] [SMulCommClass 𝕜 R R] (isometry_mul' : Isometry ⇑(ContinuousLinearMap.mul 𝕜 R)) : RegularNormedAlgebra 𝕜 R
• UniformSpace.Completion.coe_isometry 📋 Mathlib.Topology.MetricSpace.Completion
  {α : Type u} [PseudoMetricSpace α] : Isometry UniformSpace.Completion.coe'
• Isometry.completion_extension 📋 Mathlib.Topology.MetricSpace.Completion
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] [CompleteSpace β] [T0Space β] {f : α → β} (h : Isometry f) : Isometry (UniformSpace.Completion.extension f)
• Isometry.completion_map 📋 Mathlib.Topology.MetricSpace.Completion
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (h : Isometry f) : Isometry (UniformSpace.Completion.map f)
• Isometry.extensionHom 📋 Mathlib.Topology.MetricSpace.Completion
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [Ring α] [IsTopologicalRing α] [IsUniformAddGroup α] [Ring β] [PseudoMetricSpace β] [IsUniformAddGroup β] [IsTopologicalRing β] [CompleteSpace β] [T0Space β] {f : α →+* β} (h : Isometry ⇑f) : UniformSpace.Completion α →+* β
• Isometry.mapRingHom 📋 Mathlib.Topology.MetricSpace.Completion
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [Ring α] [IsTopologicalRing α] [IsUniformAddGroup α] [Ring β] [PseudoMetricSpace β] [IsUniformAddGroup β] [IsTopologicalRing β] {f : α →+* β} (h : Isometry ⇑f) : UniformSpace.Completion α →+* UniformSpace.Completion β
• Isometry.extensionHom_coe 📋 Mathlib.Topology.MetricSpace.Completion
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [Ring α] [IsTopologicalRing α] [IsUniformAddGroup α] [Ring β] [PseudoMetricSpace β] [IsUniformAddGroup β] [IsTopologicalRing β] [CompleteSpace β] [T0Space β] {f : α →+* β} (h : Isometry ⇑f) (x : α) : h.extensionHom ↑x = f x
• Isometry.isometry_mapRingHom 📋 Mathlib.Topology.MetricSpace.Completion
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [Ring α] [IsTopologicalRing α] [IsUniformAddGroup α] [Ring β] [PseudoMetricSpace β] [IsUniformAddGroup β] [IsTopologicalRing β] {f : α →+* β} (h : Isometry ⇑f) : Isometry ⇑h.mapRingHom
• Isometry.mapRingHom_coe 📋 Mathlib.Topology.MetricSpace.Completion
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [Ring α] [IsTopologicalRing α] [IsUniformAddGroup α] [Ring β] [PseudoMetricSpace β] [IsUniformAddGroup β] [IsTopologicalRing β] {f : α →+* β} (h : Isometry ⇑f) (x : α) : h.mapRingHom ↑x = ↑(f x)
• UniformSpace.Completion.isometry_mapRingHom 📋 Mathlib.Topology.MetricSpace.Completion
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [Ring α] [IsTopologicalRing α] [IsUniformAddGroup α] [Ring β] [PseudoMetricSpace β] [IsUniformAddGroup β] [IsTopologicalRing β] {f : α →+* β} (h : Isometry ⇑f) : Isometry ⇑(UniformSpace.Completion.mapRingHom f ⋯)
• AffineIsometry.isometry 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : Isometry ⇑f
• AffineIsometryEquiv.isometry 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : Isometry ⇑e
• AffineIsometryEquiv.vadd_vsub 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] {f : P → P₂} (hf : Isometry f) {p : P} {g : V → V₂} (hg : ∀ (v : V), g v = f (v +ᵥ p) -ᵥ f p) : Isometry g
• BoundedContinuousFunction.isometry_extend 📋 Mathlib.Topology.ContinuousMap.Bounded.Basic
  {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] [DiscreteTopology δ] (f : α ↪ δ) (h : BoundedContinuousFunction δ β) : Isometry fun g => BoundedContinuousFunction.extend f g h
• WithLp.prod_isometry_ofLp_infty 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  (α : Type u_2) (β : Type u_3) [PseudoEMetricSpace α] [PseudoEMetricSpace β] : Isometry WithLp.ofLp
• Isometry.withLpProdMap 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  (p : ENNReal) {α : Type u_2} {β : Type u_3} {α' : Type u_5} {β' : Type u_6} [hp : Fact (1 ≤ p)] [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace α'] [PseudoEMetricSpace β'] {f : α → α'} (hf : Isometry f) {g : β → β'} (hg : Isometry g) : Isometry (WithLp.map p (Prod.map f g))
• PiLp.isometry_ofLp_infty 📋 Mathlib.Analysis.Normed.Lp.PiLp
  {ι : Type u_2} (β : ι → Type u_4) [Fintype ι] [(i : ι) → PseudoEMetricSpace (β i)] : Isometry WithLp.ofLp
• FormalMultilinearSeries.radius_compContinuousLinearMap_eq 📋 Mathlib.Analysis.Analytic.ConvergenceRadius
  {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [Nontrivial E] (p : FormalMultilinearSeries 𝕜 F G) (u : E →L[𝕜] F) (hu_iso : Isometry ⇑u) (hu_surj : Function.Surjective ⇑u) : (p.compContinuousLinearMap u).radius = p.radius
• ContinuousMapZero.isometry_toContinuousMap 📋 Mathlib.Topology.ContinuousMap.ContinuousMapZero
  {α : Type u_1} {R : Type u_3} [TopologicalSpace α] [CompactSpace α] [Zero α] [MetricSpace R] [Zero R] : Isometry ContinuousMapZero.toContinuousMap
• AbsoluteValue.Completion.locallyCompactSpace 📋 Mathlib.Analysis.Normed.Field.WithAbs
  {K : Type u_1} [Field K] {v : AbsoluteValue K ℝ} {L : Type u_2} [NormedField L] [CompleteSpace L] {f : WithAbs v →+* L} [LocallyCompactSpace L] (h : Isometry ⇑f) : LocallyCompactSpace v.Completion
• WithAbs.isometry_map 📋 Mathlib.Analysis.Normed.Field.WithAbs
  {K : Type u_3} {L : Type u_4} [Field K] [Field L] [Algebra K L] (v : AbsoluteValue K ℝ) (w : AbsoluteValue L ℝ) [w.LiesOver v] : Isometry ⇑(WithAbs.map v w (algebraMap K L))
• Isometry.isCover_image_iff 📋 Mathlib.Topology.MetricSpace.Cover
  {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {ε : NNReal} {s : Set X} {f : X → Y} (hf : Isometry f) (C : Set X) : Metric.IsCover ε (f '' s) (f '' C) ↔ Metric.IsCover ε s C
• Unitization.isometry_inr 📋 Mathlib.Analysis.Normed.Algebra.Unitization
  {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A] : Isometry Unitization.inr
• ContinuousLinearMap.isometry_mul_flip 📋 Mathlib.Analysis.CStarAlgebra.Unitization
  (𝕜 : Type u_1) (E : Type u_2) [NontriviallyNormedField 𝕜] [NonUnitalNormedRing E] [StarRing E] [NormedStarGroup E] [NormedSpace 𝕜 E] [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] [RegularNormedAlgebra 𝕜 E] : Isometry ⇑(ContinuousLinearMap.mul 𝕜 E).flip
• WithLp.unitization_isometry_inr 📋 Mathlib.Analysis.Normed.Algebra.UnitizationL1
  {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] : Isometry fun x => WithLp.toLp 1 ↑x
• SpectrumRestricts.isometric_cfc 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
  {R : Type u_1} {S : Type u_2} {A : Type u_3} {p q : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [Semifield S] [StarRing S] [MetricSpace S] [IsTopologicalSemiring S] [ContinuousStar S] [Ring A] [StarRing A] [Algebra S A] [Algebra R S] [Algebra R A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] [MetricSpace A] [IsometricContinuousFunctionalCalculus S A q] [CompleteSpace R] [ContinuousMap.UniqueHom R A] (f : C(S, R)) (halg : Isometry ⇑(algebraMap R S)) (h0 : p 0) (h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f) : IsometricContinuousFunctionalCalculus R A p
• IsometricContinuousFunctionalCalculus.mk 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
  {R : Type u_1} {A : Type u_2} {p : outParam (A → Prop)} [CommSemiring R] [StarRing R] [MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [Ring A] [StarRing A] [MetricSpace A] [Algebra R A] [toContinuousFunctionalCalculus : ContinuousFunctionalCalculus R A p] (isometric : ∀ (a : A) (ha : p a), Isometry ⇑(cfcHom ha)) : IsometricContinuousFunctionalCalculus R A p
• IsometricContinuousFunctionalCalculus.isometric 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
  {R : Type u_1} {A : Type u_2} {p : outParam (A → Prop)} {inst✝ : CommSemiring R} {inst✝¹ : StarRing R} {inst✝² : MetricSpace R} {inst✝³ : IsTopologicalSemiring R} {inst✝⁴ : ContinuousStar R} {inst✝⁵ : Ring A} {inst✝⁶ : StarRing A} {inst✝⁷ : MetricSpace A} {inst✝⁸ : Algebra R A} [self : IsometricContinuousFunctionalCalculus R A p] (a : A) (ha : p a) : Isometry ⇑(cfcHom ha)
• isometry_cfcHom 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
  {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] [MetricSpace A] [Algebra R A] [IsometricContinuousFunctionalCalculus R A p] (a : A) (ha : p a := by cfc_tac) : Isometry ⇑(cfcHom ⋯)
• QuasispectrumRestricts.isometric_cfc 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
  {R : Type u_1} {S : Type u_2} {A : Type u_3} {p q : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [Field S] [StarRing S] [MetricSpace S] [IsTopologicalRing S] [ContinuousStar S] [NonUnitalRing A] [StarRing A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Algebra R S] [Module R A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] [IsScalarTower R A A] [SMulCommClass R A A] [MetricSpace A] [NonUnitalIsometricContinuousFunctionalCalculus S A q] [CompleteSpace R] [ContinuousMapZero.UniqueHom R A] (f : C(S, R)) (halg : Isometry ⇑(algebraMap R S)) (h0 : p 0) (h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f) : NonUnitalIsometricContinuousFunctionalCalculus R A p
• NonUnitalIsometricContinuousFunctionalCalculus.mk 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
  {R : Type u_1} {A : Type u_2} {p : outParam (A → Prop)} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [MetricSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [toNonUnitalContinuousFunctionalCalculus : NonUnitalContinuousFunctionalCalculus R A p] (isometric : ∀ (a : A) (ha : p a), Isometry ⇑(cfcₙHom ha)) : NonUnitalIsometricContinuousFunctionalCalculus R A p
• NonUnitalIsometricContinuousFunctionalCalculus.isometric 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
  {R : Type u_1} {A : Type u_2} {p : outParam (A → Prop)} {inst✝ : CommSemiring R} {inst✝¹ : Nontrivial R} {inst✝² : StarRing R} {inst✝³ : MetricSpace R} {inst✝⁴ : IsTopologicalSemiring R} {inst✝⁵ : ContinuousStar R} {inst✝⁶ : NonUnitalRing A} {inst✝⁷ : StarRing A} {inst✝⁸ : MetricSpace A} {inst✝⁹ : Module R A} {inst✝¹⁰ : IsScalarTower R A A} {inst✝¹¹ : SMulCommClass R A A} [self : NonUnitalIsometricContinuousFunctionalCalculus R A p] (a : A) (ha : p a) : Isometry ⇑(cfcₙHom ha)
• isometry_cfcₙHom 📋 Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
  {R : Type u_1} {A : Type u_2} {p : outParam (A → Prop)} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A] [MetricSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalIsometricContinuousFunctionalCalculus R A p] (a : A) (ha : p a := by cfc_tac) : Isometry ⇑(cfcₙHom ⋯)
• StarAlgEquiv.isometry 📋 Mathlib.Analysis.CStarAlgebra.Spectrum
  {F : Type u_1} {A : Type u_2} {B : Type u_3} [NonUnitalCStarAlgebra A] [NonUnitalCStarAlgebra B] [EquivLike F A B] [NonUnitalAlgEquivClass F ℂ A B] [StarHomClass F A B] (φ : F) : Isometry ⇑φ
• ZeroAtInftyContinuousMap.isometry_toBCF 📋 Mathlib.Topology.ContinuousMap.ZeroAtInfty
  {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] : Isometry ZeroAtInftyContinuousMap.toBCF
• gelfandTransform_isometry 📋 Mathlib.Analysis.CStarAlgebra.GelfandDuality
  (A : Type u_1) [CommCStarAlgebra A] : Isometry ⇑(WeakDual.gelfandTransform ℂ A)
• UniformFun.isometry_ofFun_boundedContinuousFunction 📋 Mathlib.Topology.MetricSpace.UniformConvergence
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace β] [TopologicalSpace α] : Isometry (⇑UniformFun.ofFun ∘ DFunLike.coe)
• UniformFun.isometry_ofFun_continuousMap 📋 Mathlib.Topology.MetricSpace.UniformConvergence
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace β] [TopologicalSpace α] [CompactSpace α] : Isometry (⇑UniformFun.ofFun ∘ DFunLike.coe)
• UniformOnFun.isometry_restrict 📋 Mathlib.Topology.MetricSpace.UniformConvergence
  {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace β] (s : Set α) : Isometry (⇑UniformFun.ofFun ∘ s.domRestrict ∘ ⇑(UniformOnFun.toFun {s}))
• ContinuousMap.isometry_realToRCLike 📋 Mathlib.Analysis.RCLike.ContinuousMap
  (X : Type u_1) (𝕜 : Type u_2) [TopologicalSpace X] [RCLike 𝕜] [CompactSpace X] : Isometry (ContinuousMap.realToRCLike 𝕜)
• NonUnitalStarAlgHom.isometry 📋 Mathlib.Analysis.CStarAlgebra.Hom
  {F : Type u_1} {A : Type u_2} {B : Type u_3} [NonUnitalCStarAlgebra A] [NonUnitalCStarAlgebra B] [FunLike F A B] [NonUnitalAlgHomClass F ℂ A B] [StarHomClass F A B] (φ : F) (hφ : Function.Injective ⇑φ) : Isometry ⇑φ
• ContinuousLinearMap.isometry_iff_adjoint_comp_self 📋 Mathlib.Analysis.InnerProductSpace.Adjoint
  {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] [CompleteSpace H] {K : Type u_6} [NormedAddCommGroup K] [InnerProductSpace 𝕜 K] [CompleteSpace K] (u : H →L[𝕜] K) : Isometry ⇑u ↔ ContinuousLinearMap.adjoint u ∘SL u = 1
• lp.isometry_single 📋 Mathlib.Analysis.Normed.Lp.lpSpace
  {α : Type u_3} {E : α → Type u_4} {p : ENNReal} [(i : α) → NormedAddCommGroup (E i)] [DecidableEq α] [Fact (1 ≤ p)] (i : α) : Isometry (lp.single p i)
• UpperHalfPlane.isometry_vertical_line 📋 Mathlib.Analysis.Complex.UpperHalfPlane.Metric
  (a : ℝ) : Isometry fun y => { coe := { re := a, im := Real.exp y }, coe_im_pos := ⋯ }
• UpperHalfPlane.isometry_real_vadd 📋 Mathlib.Analysis.Complex.UpperHalfPlane.Metric
  (a : ℝ) : Isometry fun x => a +ᵥ x
• UpperHalfPlane.isometry_pos_mul 📋 Mathlib.Analysis.Complex.UpperHalfPlane.Metric
  (a : { x // 0 < x }) : Isometry fun x => a • x
• Isometry.affineIsometryOfStrictConvexSpace 📋 Mathlib.Analysis.Convex.StrictConvexBetween
  {E : Type u_3} {F : Type u_4} {PE : Type u_5} {PF : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] [StrictConvexSpace ℝ E] [MetricSpace PE] [MetricSpace PF] [NormedAddTorsor E PE] [NormedAddTorsor F PF] {f : PF → PE} (hi : Isometry f) : PF →ᵃⁱ[ℝ] PE
• Isometry.coe_affineIsometryOfStrictConvexSpace 📋 Mathlib.Analysis.Convex.StrictConvexBetween
  {E : Type u_3} {F : Type u_4} {PE : Type u_5} {PF : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] [StrictConvexSpace ℝ E] [MetricSpace PE] [MetricSpace PF] [NormedAddTorsor E PE] [NormedAddTorsor F PF] {f : PF → PE} (hi : Isometry f) : ⇑hi.affineIsometryOfStrictConvexSpace = f
• Isometry.affineIsometryOfStrictConvexSpace_apply 📋 Mathlib.Analysis.Convex.StrictConvexBetween
  {E : Type u_3} {F : Type u_4} {PE : Type u_5} {PF : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] [StrictConvexSpace ℝ E] [MetricSpace PE] [MetricSpace PF] [NormedAddTorsor E PE] [NormedAddTorsor F PF] {f : PF → PE} (hi : Isometry f) (p : PF) : hi.affineIsometryOfStrictConvexSpace p = f p
• Isometry.map_hausdorffMeasure 📋 Mathlib.MeasureTheory.Measure.Hausdorff
  {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {d : ℝ} (hf : Isometry f) (hd : 0 ≤ d ∨ Function.Surjective f) : MeasureTheory.Measure.map f (MeasureTheory.Measure.hausdorffMeasure d) = (MeasureTheory.Measure.hausdorffMeasure d).restrict (Set.range f)
• Isometry.hausdorffMeasure_image 📋 Mathlib.MeasureTheory.Measure.Hausdorff
  {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {d : ℝ} (hf : Isometry f) (hd : 0 ≤ d ∨ Function.Surjective f) (s : Set X) : (MeasureTheory.Measure.hausdorffMeasure d) (f '' s) = (MeasureTheory.Measure.hausdorffMeasure d) s
• Isometry.hausdorffMeasure_preimage 📋 Mathlib.MeasureTheory.Measure.Hausdorff
  {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {d : ℝ} (hf : Isometry f) (hd : 0 ≤ d ∨ Function.Surjective f) (s : Set Y) : (MeasureTheory.Measure.hausdorffMeasure d) (f ⁻¹' s) = (MeasureTheory.Measure.hausdorffMeasure d) (s ∩ Set.range f)
• MeasureTheory.OuterMeasure.isometry_comap_mkMetric 📋 Mathlib.MeasureTheory.Measure.Hausdorff
  {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] (m : ENNReal → ENNReal) {f : X → Y} (hf : Isometry f) (H : Monotone m ∨ Function.Surjective f) : (MeasureTheory.OuterMeasure.comap f) (MeasureTheory.OuterMeasure.mkMetric m) = MeasureTheory.OuterMeasure.mkMetric m
• MeasureTheory.OuterMeasure.isometry_map_mkMetric 📋 Mathlib.MeasureTheory.Measure.Hausdorff
  {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] (m : ENNReal → ENNReal) {f : X → Y} (hf : Isometry f) (H : Monotone m ∨ Function.Surjective f) : (MeasureTheory.OuterMeasure.map f) (MeasureTheory.OuterMeasure.mkMetric m) = (MeasureTheory.OuterMeasure.restrict (Set.range f)) (MeasureTheory.OuterMeasure.mkMetric m)
• Isometry.map_euclideanHausdorffMeasure 📋 Mathlib.Geometry.Euclidean.Volume.Measure
  {X : Type u_1} {Y : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] [EMetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {d : ℕ} (hf : Isometry f) : MeasureTheory.Measure.map f (MeasureTheory.Measure.euclideanHausdorffMeasure d) = (MeasureTheory.Measure.euclideanHausdorffMeasure d).restrict (Set.range f)
• Isometry.euclideanHausdorffMeasure_image 📋 Mathlib.Geometry.Euclidean.Volume.Measure
  {X : Type u_1} {Y : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] [EMetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {d : ℕ} (hf : Isometry f) (s : Set X) : (MeasureTheory.Measure.euclideanHausdorffMeasure d) (f '' s) = (MeasureTheory.Measure.euclideanHausdorffMeasure d) s
• Isometry.euclideanHausdorffMeasure_preimage 📋 Mathlib.Geometry.Euclidean.Volume.Measure
  {X : Type u_1} {Y : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] [EMetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] {f : X → Y} {d : ℕ} (hf : Isometry f) (s : Set Y) : (MeasureTheory.Measure.euclideanHausdorffMeasure d) (f ⁻¹' s) = (MeasureTheory.Measure.euclideanHausdorffMeasure d) (s ∩ Set.range f)
• Isometry.dimH_image 📋 Mathlib.Topology.MetricSpace.HausdorffDimension
  {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] {f : X → Y} (hf : Isometry f) (s : Set X) : dimH (f '' s) = dimH s
• SemiNormedGrp₁.iso_isometry 📋 Mathlib.Analysis.Normed.Group.SemiNormedGrp
  {V W : SemiNormedGrp₁} (i : V ≅ W) : Isometry ⇑↑(SemiNormedGrp₁.Hom.hom i.hom)
• SemiNormedGrp.iso_isometry_of_normNoninc 📋 Mathlib.Analysis.Normed.Group.SemiNormedGrp
  {V W : SemiNormedGrp} (i : V ≅ W) (h1 : (SemiNormedGrp.Hom.hom i.hom).NormNoninc) (h2 : (SemiNormedGrp.Hom.hom i.inv).NormNoninc) : Isometry ⇑(CategoryTheory.ConcreteCategory.hom i.hom)
• Polynomial.leadingCoeff_le_mapMahlerMeasure 📋 Mathlib.Analysis.Polynomial.MahlerMeasure
  {A : Type u_2} [NormedRing A] (p : Polynomial A) (v : A →+* ℂ) (hv : Isometry ⇑v) : ‖p.leadingCoeff‖ ≤ p.mapMahlerMeasure v
• Polynomial.mapMahlerMeasure_le_sum_norm_coeff 📋 Mathlib.Analysis.Polynomial.MahlerMeasure
  {A : Type u_2} [NormedRing A] (p : Polynomial A) (v : A →+* ℂ) (hv : Isometry ⇑v) : p.mapMahlerMeasure v ≤ p.sum fun x a => ‖a‖
• Polynomial.Monic.one_le_mapMahlerMeasure 📋 Mathlib.Analysis.Polynomial.MahlerMeasure
  {A : Type u_2} [NormedRing A] {p : Polynomial A} (v : A →+* ℂ) [NormOneClass A] (hv : Isometry ⇑v) (hp : p.Monic) : 1 ≤ p.mapMahlerMeasure v
• Polynomial.norm_coeff_le_choose_mul_mapMahlerMeasure 📋 Mathlib.Analysis.Polynomial.MahlerMeasure
  {A : Type u_2} [NormedRing A] (v : A →+* ℂ) (hv : Isometry ⇑v) (n : ℕ) (p : Polynomial A) : ‖p.coeff n‖ ≤ ↑(p.natDegree.choose n) * p.mapMahlerMeasure v

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:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. 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

4. 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.

5. 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 69fae59