ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AffineIsometryEquiv.bijective
protected theorem bijective : Bijective e 
Arguments: {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)
Goal: Function.Bijective ⇑e
-- AffineIsometryEquiv.injective
protected theorem injective : Injective e 
Arguments: {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)
Goal: Function.Injective ⇑e
-- AffineIsometryEquiv.surjective
protected theorem surjective : Surjective e 
Arguments: {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)
Goal: Function.Surjective ⇑e
-- AffineIsometryEquiv.map_eq_iff
theorem map_eq_iff {x y : P} : e x = e y ↔ x = y 
Arguments: {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) {x y : P}
Goal: e x = e y ↔ x = y
-- AffineIsometryEquiv.map_ne
theorem map_ne {x y : P} (h : x ≠ y) : e x ≠ e y 
Arguments: {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) {x y : P} (h : x ≠ y)
Goal: e x ≠ e y
-- AffineIsometryEquiv.lipschitz
protected theorem lipschitz : LipschitzWith 1 e 
Arguments: {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)
Goal: LipschitzWith 1 ⇑e
-- AffineIsometryEquiv.antilipschitz
protected theorem antilipschitz : AntilipschitzWith 1 e 
Arguments: {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)
Goal: AntilipschitzWith 1 ⇑e
-- AffineIsometryEquiv.ediam_image
@[simp]
theorem ediam_image (s : Set P) : EMetric.diam (e '' s) = EMetric.diam s 
Arguments: {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) (s : Set P)
Goal: EMetric.diam (⇑e '' s) = EMetric.diam s
-- AffineIsometryEquiv.diam_image
@[simp]
theorem diam_image (s : Set P) : Metric.diam (e '' s) = Metric.diam s 
Arguments: {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) (s : Set P)
Goal: Metric.diam (⇑e '' s) = Metric.diam s
-- AffineIsometryEquiv.comp_continuousOn_iff
@[simp]
theorem comp_continuousOn_iff {f : α → P} {s : Set α} : ContinuousOn (e ∘ f) s ↔ ContinuousOn f s 
Arguments: {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) {α : Type u_14} [TopologicalSpace α] {f : α → P} {s : Set α}
Goal: ContinuousOn (⇑e ∘ f) s ↔ ContinuousOn f s
-- AffineIsometryEquiv.comp_continuous_iff
@[simp]
theorem comp_continuous_iff {f : α → P} : Continuous (e ∘ f) ↔ Continuous f 
Arguments: {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) {α : Type u_14} [TopologicalSpace α] {f : α → P}
Goal: Continuous (⇑e ∘ f) ↔ Continuous f
-- AffineIsometryEquiv.coe_vaddConst
@[simp]
theorem coe_vaddConst (p : P) : ⇑(vaddConst 𝕜 p) = fun v => v +ᵥ p 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P)
Goal: ⇑(AffineIsometryEquiv.vaddConst 𝕜 p) = fun v => v +ᵥ p
-- AffineIsometryEquiv.coe_vaddConst'
@[simp]
theorem coe_vaddConst' (p : P) : ↑(AffineEquiv.vaddConst 𝕜 p) = fun v => v +ᵥ p 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P)
Goal: ⇑(AffineEquiv.vaddConst 𝕜 p) = fun v => v +ᵥ p
-- AffineIsometryEquiv.coe_vaddConst_symm
@[simp]
theorem coe_vaddConst_symm (p : P) : ⇑(vaddConst 𝕜 p).symm = fun p' => p' -ᵥ p 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P)
Goal: ⇑(AffineIsometryEquiv.vaddConst 𝕜 p).symm = fun p' => p' -ᵥ p
-- AffineIsometryEquiv.vaddConst_toAffineEquiv
@[simp]
theorem vaddConst_toAffineEquiv (p : P) :
    (vaddConst 𝕜 p).toAffineEquiv = AffineEquiv.vaddConst 𝕜 p 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P)
Goal: (AffineIsometryEquiv.vaddConst 𝕜 p).toAffineEquiv = AffineEquiv.vaddConst 𝕜 p
-- AffineIsometryEquiv.coe_constVSub
@[simp]
theorem coe_constVSub (p : P) : ⇑(constVSub 𝕜 p) = (p -ᵥ ·) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P)
Goal: ⇑(AffineIsometryEquiv.constVSub 𝕜 p) = fun x => p -ᵥ x
-- AffineIsometryEquiv.symm_constVSub
@[simp]
theorem symm_constVSub (p : P) :
    (constVSub 𝕜 p).symm =
      (LinearIsometryEquiv.neg 𝕜).toAffineIsometryEquiv.trans (vaddConst 𝕜 p) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P)
Goal: (AffineIsometryEquiv.constVSub 𝕜 p).symm =
  (LinearIsometryEquiv.neg 𝕜).toAffineIsometryEquiv.trans (AffineIsometryEquiv.vaddConst 𝕜 p)
-- AffineIsometryEquiv.coe_constVAdd
@[simp]
theorem coe_constVAdd (v : V) : ⇑(constVAdd 𝕜 P v : P ≃ᵃⁱ[𝕜] P) = (v +ᵥ ·) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V)
Goal: ⇑(AffineIsometryEquiv.constVAdd 𝕜 P v) = fun x => v +ᵥ x
-- AffineIsometryEquiv.constVAdd_zero
@[simp]
theorem constVAdd_zero : constVAdd 𝕜 P (0 : V) = refl 𝕜 P 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P]
Goal: AffineIsometryEquiv.constVAdd 𝕜 P 0 = AffineIsometryEquiv.refl 𝕜 P
-- AffineIsometryEquiv.vadd_vsub
theorem vadd_vsub {f : P → P₂} (hf : Isometry f) {p : P} {g : V → V₂}
    (hg : ∀ v, g v = f (v +ᵥ p) -ᵥ f p) : Isometry g 
Arguments: {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)
Goal: Isometry g
-- AffineIsometryEquiv.pointReflection_apply
theorem pointReflection_apply (x y : P) : (pointReflection 𝕜 x) y = x -ᵥ y +ᵥ x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P)
Goal: (AffineIsometryEquiv.pointReflection 𝕜 x) y = x -ᵥ y +ᵥ x
-- AffineIsometryEquiv.pointReflection_toAffineEquiv
@[simp]
theorem pointReflection_toAffineEquiv (x : P) :
    (pointReflection 𝕜 x).toAffineEquiv = AffineEquiv.pointReflection 𝕜 x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P)
Goal: (AffineIsometryEquiv.pointReflection 𝕜 x).toAffineEquiv = AffineEquiv.pointReflection 𝕜 x
-- AffineIsometryEquiv.pointReflection_self
@[simp]
theorem pointReflection_self (x : P) : pointReflection 𝕜 x x = x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P)
Goal: (AffineIsometryEquiv.pointReflection 𝕜 x) x = x
-- AffineIsometryEquiv.pointReflection_involutive
theorem pointReflection_involutive (x : P) : Function.Involutive (pointReflection 𝕜 x) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P)
Goal: Function.Involutive ⇑(AffineIsometryEquiv.pointReflection 𝕜 x)
-- AffineIsometryEquiv.pointReflection_symm
@[simp]
theorem pointReflection_symm (x : P) : (pointReflection 𝕜 x).symm = pointReflection 𝕜 x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P)
Goal: (AffineIsometryEquiv.pointReflection 𝕜 x).symm = AffineIsometryEquiv.pointReflection 𝕜 x
-- AffineIsometryEquiv.dist_pointReflection_fixed
@[simp]
theorem dist_pointReflection_fixed (x y : P) : dist (pointReflection 𝕜 x y) x = dist y x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P)
Goal: dist ((AffineIsometryEquiv.pointReflection 𝕜 x) y) x = dist y x
-- AffineIsometryEquiv.dist_pointReflection_self'
theorem dist_pointReflection_self' (x y : P) :
    dist (pointReflection 𝕜 x y) y = ‖2 • (x -ᵥ y)‖ 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P)
Goal: dist ((AffineIsometryEquiv.pointReflection 𝕜 x) y) y = ‖2 • (x -ᵥ y)‖
-- AffineIsometryEquiv.dist_pointReflection_self
theorem dist_pointReflection_self (x y : P) :
    dist (pointReflection 𝕜 x y) y = ‖(2 : 𝕜)‖ * dist x y 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P)
Goal: dist ((AffineIsometryEquiv.pointReflection 𝕜 x) y) y = ‖2‖ * dist x y
-- AffineIsometryEquiv.pointReflection_fixed_iff
theorem pointReflection_fixed_iff [Invertible (2 : 𝕜)] {x y : P} :
    pointReflection 𝕜 x y = y ↔ y = x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [Invertible 2] {x y : P}
Goal: (AffineIsometryEquiv.pointReflection 𝕜 x) y = y ↔ y = x
-- AffineIsometryEquiv.dist_pointReflection_self_real
theorem dist_pointReflection_self_real (x y : P) :
    dist (pointReflection ℝ x y) y = 2 * dist x y 
Arguments: {V : Type u_2} {P : Type u_10} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [NormedSpace ℝ V] (x y : P)
Goal: dist ((AffineIsometryEquiv.pointReflection ℝ x) y) y = 2 * dist x y
-- AffineIsometryEquiv.pointReflection_midpoint_left
@[simp]
theorem pointReflection_midpoint_left (x y : P) : pointReflection ℝ (midpoint ℝ x y) x = y 
Arguments: {V : Type u_2} {P : Type u_10} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [NormedSpace ℝ V] (x y : P)
Goal: (AffineIsometryEquiv.pointReflection ℝ (midpoint ℝ x y)) x = y
-- AffineIsometryEquiv.pointReflection_midpoint_right
@[simp]
theorem pointReflection_midpoint_right (x y : P) : pointReflection ℝ (midpoint ℝ x y) y = x 
Arguments: {V : Type u_2} {P : Type u_10} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [NormedSpace ℝ V] (x y : P)
Goal: (AffineIsometryEquiv.pointReflection ℝ (midpoint ℝ x y)) y = x
-- AffineMap.continuous_linear_iff
theorem AffineMap.continuous_linear_iff {f : P →ᵃ[𝕜] P₂} : Continuous f.linear ↔ Continuous f 
Arguments: {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 →ᵃ[𝕜] P₂}
Goal: Continuous ⇑f.linear ↔ Continuous ⇑f
-- AffineMap.isOpenMap_linear_iff
theorem AffineMap.isOpenMap_linear_iff {f : P →ᵃ[𝕜] P₂} : IsOpenMap f.linear ↔ IsOpenMap f 
Arguments: {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 →ᵃ[𝕜] P₂}
Goal: IsOpenMap ⇑f.linear ↔ IsOpenMap ⇑f
-- AffineSubspace.linear_equivMapOfInjective
@[simps linear, simps! toFun]
noncomputable def equivMapOfInjective (E : AffineSubspace 𝕜 P₁) [Nonempty E] (φ : P₁ →ᵃ[𝕜] P₂)
    (hφ : Function.Injective φ) : E ≃ᵃ[𝕜] E.map φ 
Arguments: {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [NormedSpace 𝕜 V₁] [PseudoMetricSpace P₁] [NormedAddTorsor V₁ P₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : Function.Injective ⇑φ)
Goal: (E.equivMapOfInjective φ hφ).linear =
  (Submodule.equivMapOfInjective φ.linear ⋯ E.direction).trans
    (LinearEquiv.ofEq (Submodule.map φ.linear E.direction) (AffineSubspace.map φ E).direction ⋯)
-- AffineSubspace.equivMapOfInjective_toFun
@[simps linear, simps! toFun]
noncomputable def equivMapOfInjective (E : AffineSubspace 𝕜 P₁) [Nonempty E] (φ : P₁ →ᵃ[𝕜] P₂)
    (hφ : Function.Injective φ) : E ≃ᵃ[𝕜] E.map φ 
Arguments: {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [NormedSpace 𝕜 V₁] [PseudoMetricSpace P₁] [NormedAddTorsor V₁ P₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : Function.Injective ⇑φ) (p : ↑↑E)
Goal: (E.equivMapOfInjective φ hφ) p = ⟨φ ↑p, ⋯⟩
-- AffineSubspace.isometryEquivMap.apply_symm_apply
@[simp]
theorem isometryEquivMap.apply_symm_apply {E : AffineSubspace 𝕜 P₁'} [Nonempty E]
    {φ : P₁' →ᵃⁱ[𝕜] P₂} (x : E.map φ.toAffineMap) : φ ((E.isometryEquivMap φ).symm x) = x 
Arguments: {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] {E : AffineSubspace 𝕜 P₁'} [Nonempty ↥E] {φ : P₁' →ᵃⁱ[𝕜] P₂} (x : ↥(AffineSubspace.map φ.toAffineMap E))
Goal: φ ↑((AffineSubspace.isometryEquivMap φ E).symm x) = ↑x
-- AffineSubspace.isometryEquivMap.coe_apply
@[simp]
theorem isometryEquivMap.coe_apply (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁') [Nonempty E]
    (g : E) : ↑(E.isometryEquivMap φ g) = φ g 
Arguments: {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁') [Nonempty ↥E] (g : ↥E)
Goal: ↑((AffineSubspace.isometryEquivMap φ E) g) = φ ↑g
-- AffineSubspace.isometryEquivMap.toAffineMap_eq
@[simp]
theorem isometryEquivMap.toAffineMap_eq (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁')
    [Nonempty E] :
    (E.isometryEquivMap φ).toAffineMap = E.equivMapOfInjective φ.toAffineMap φ.injective 
Arguments: {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁') [Nonempty ↥E]
Goal: ↑(AffineSubspace.isometryEquivMap φ E).toAffineEquiv = ↑(E.equivMapOfInjective φ.toAffineMap ⋯)
-- Complex.summable_one_div_nat_cpow
lemma Complex.summable_one_div_nat_cpow {p : ℂ} :
    Summable (fun n : ℕ ↦ 1 / (n : ℂ) ^ p) ↔ 1 < re p 
Arguments: {p : ℂ}
Goal: (Summable fun n => 1 / ↑n ^ p) ↔ 1 < p.re
-- ediam_smul_le
theorem ediam_smul_le (c : 𝕜) (s : Set E) : EMetric.diam (c • s) ≤ ‖c‖₊ • EMetric.diam s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedAddCommGroup 𝕜] [SeminormedAddCommGroup E] [SMulZeroClass 𝕜 E] [BoundedSMul 𝕜 E] (c : 𝕜) (s : Set E)
Goal: EMetric.diam (c • s) ≤ ‖c‖₊ • EMetric.diam s
-- ediam_smul₀
theorem ediam_smul₀ (c : 𝕜) (s : Set E) : EMetric.diam (c • s) = ‖c‖₊ • EMetric.diam s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedDivisionRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] (c : 𝕜) (s : Set E)
Goal: EMetric.diam (c • s) = ‖c‖₊ • EMetric.diam s
-- diam_smul₀
theorem diam_smul₀ (c : 𝕜) (x : Set E) : diam (c • x) = ‖c‖ * diam x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedDivisionRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] (c : 𝕜) (x : Set E)
Goal: Metric.diam (c • x) = ‖c‖ * Metric.diam x
-- infEdist_smul₀
theorem infEdist_smul₀ {c : 𝕜} (hc : c ≠ 0) (s : Set E) (x : E) :
    EMetric.infEdist (c • x) (c • s) = ‖c‖₊ • EMetric.infEdist x s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedDivisionRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] {c : 𝕜} (hc : c ≠ 0) (s : Set E) (x : E)
Goal: EMetric.infEdist (c • x) (c • s) = ‖c‖₊ • EMetric.infEdist x s
-- infDist_smul₀
theorem infDist_smul₀ {c : 𝕜} (hc : c ≠ 0) (s : Set E) (x : E) :
    Metric.infDist (c • x) (c • s) = ‖c‖ * Metric.infDist x s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedDivisionRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] {c : 𝕜} (hc : c ≠ 0) (s : Set E) (x : E)
Goal: Metric.infDist (c • x) (c • s) = ‖c‖ * Metric.infDist x s
-- smul_ball
theorem smul_ball {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) : c • ball x r = ball (c • x) (‖c‖ * r) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ)
Goal: c • Metric.ball x r = Metric.ball (c • x) (‖c‖ * r)
-- smul_unitBall
theorem smul_unitBall {c : 𝕜} (hc : c ≠ 0) : c • ball (0 : E) (1 : ℝ) = ball (0 : E) ‖c‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {c : 𝕜} (hc : c ≠ 0)
Goal: c • Metric.ball 0 1 = Metric.ball 0 ‖c‖
-- smul_sphere'
theorem smul_sphere' {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) :
    c • sphere x r = sphere (c • x) (‖c‖ * r) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ)
Goal: c • Metric.sphere x r = Metric.sphere (c • x) (‖c‖ * r)
-- smul_closedBall'
theorem smul_closedBall' {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) :
    c • closedBall x r = closedBall (c • x) (‖c‖ * r) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ)
Goal: c • Metric.closedBall x r = Metric.closedBall (c • x) (‖c‖ * r)
-- set_smul_sphere_zero
theorem set_smul_sphere_zero {s : Set 𝕜} (hs : 0 ∉ s) (r : ℝ) :
    s • sphere (0 : E) r = (‖·‖) ⁻¹' ((‖·‖ * r) '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set 𝕜} (hs : 0 ∉ s) (r : ℝ)
Goal: s • Metric.sphere 0 r = (fun x => ‖x‖) ⁻¹' ((fun x => ‖x‖ * r) '' s)
-- Bornology.IsBounded.smul₀
theorem Bornology.IsBounded.smul₀ {s : Set E} (hs : IsBounded s) (c : 𝕜) : IsBounded (c • s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} (hs : Bornology.IsBounded s) (c : 𝕜)
Goal: Bornology.IsBounded (c • s)
-- eventually_singleton_add_smul_subset
theorem eventually_singleton_add_smul_subset {x : E} {s : Set E} (hs : Bornology.IsBounded s)
    {u : Set E} (hu : u ∈ 𝓝 x) : ∀ᶠ r in 𝓝 (0 : 𝕜), {x} + r • s ⊆ u 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} (hs : Bornology.IsBounded s) {u : Set E} (hu : u ∈ 𝓝 x)
Goal: ∀ᶠ (r : 𝕜) in 𝓝 0, {x} + r • s ⊆ u
-- smul_unitBall_of_pos
theorem smul_unitBall_of_pos {r : ℝ} (hr : 0 < r) : r • ball (0 : E) 1 = ball (0 : E) r 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {r : ℝ} (hr : 0 < r)
Goal: r • Metric.ball 0 1 = Metric.ball 0 r
-- Ioo_smul_sphere_zero
lemma Ioo_smul_sphere_zero {a b r : ℝ} (ha : 0 ≤ a) (hr : 0 < r) :
    Ioo a b • sphere (0 : E) r = ball 0 (b * r) \ closedBall 0 (a * r) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {a b r : ℝ} (ha : 0 ≤ a) (hr : 0 < r)
Goal: Set.Ioo a b • Metric.sphere 0 r = Metric.ball 0 (b * r) \ Metric.closedBall 0 (a * r)
-- exists_dist_eq
theorem exists_dist_eq (x z : E) {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
    ∃ y, dist x y = b * dist x z ∧ dist y z = a * dist x z 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x z : E) {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1)
Goal: ∃ y, dist x y = b * dist x z ∧ dist y z = a * dist x z
-- exists_dist_le_le
theorem exists_dist_le_le (hδ : 0 ≤ δ) (hε : 0 ≤ ε) (h : dist x z ≤ ε + δ) :
    ∃ y, dist x y ≤ δ ∧ dist y z ≤ ε 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x z : E} {δ ε : ℝ} (hδ : 0 ≤ δ) (hε : 0 ≤ ε) (h : dist x z ≤ ε + δ)
Goal: ∃ y, dist x y ≤ δ ∧ dist y z ≤ ε
-- exists_dist_le_lt
theorem exists_dist_le_lt (hδ : 0 ≤ δ) (hε : 0 < ε) (h : dist x z < ε + δ) :
    ∃ y, dist x y ≤ δ ∧ dist y z < ε 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x z : E} {δ ε : ℝ} (hδ : 0 ≤ δ) (hε : 0 < ε) (h : dist x z < ε + δ)
Goal: ∃ y, dist x y ≤ δ ∧ dist y z < ε
-- exists_dist_lt_le
theorem exists_dist_lt_le (hδ : 0 < δ) (hε : 0 ≤ ε) (h : dist x z < ε + δ) :
    ∃ y, dist x y < δ ∧ dist y z ≤ ε 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x z : E} {δ ε : ℝ} (hδ : 0 < δ) (hε : 0 ≤ ε) (h : dist x z < ε + δ)
Goal: ∃ y, dist x y < δ ∧ dist y z ≤ ε
-- exists_dist_lt_lt
theorem exists_dist_lt_lt (hδ : 0 < δ) (hε : 0 < ε) (h : dist x z < ε + δ) :
    ∃ y, dist x y < δ ∧ dist y z < ε 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x z : E} {δ ε : ℝ} (hδ : 0 < δ) (hε : 0 < ε) (h : dist x z < ε + δ)
Goal: ∃ y, dist x y < δ ∧ dist y z < ε
-- disjoint_ball_ball_iff
theorem disjoint_ball_ball_iff (hδ : 0 < δ) (hε : 0 < ε) :
    Disjoint (ball x δ) (ball y ε) ↔ δ + ε ≤ dist x y 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x y : E} {δ ε : ℝ} (hδ : 0 < δ) (hε : 0 < ε)
Goal: Disjoint (Metric.ball x δ) (Metric.ball y ε) ↔ δ + ε ≤ dist x y