ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AffineIsometry.id_comp
@[simp]
theorem id_comp : (id : P₂ →ᵃⁱ[𝕜] P₂).comp f = 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 →ᵃⁱ[𝕜] P₂)
Goal: AffineIsometry.id.comp f = f
-- AffineIsometry.comp_id
@[simp]
theorem comp_id : f.comp id = 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 →ᵃⁱ[𝕜] P₂)
Goal: f.comp AffineIsometry.id = f
-- AffineIsometry.comp_assoc
theorem comp_assoc (f : P₃ →ᵃⁱ[𝕜] P₄) (g : P₂ →ᵃⁱ[𝕜] P₃) (h : P →ᵃⁱ[𝕜] P₂) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {V₄ : Type u_7} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} {P₄ : Type u_13} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P₃] [NormedAddTorsor V₃ P₃] [SeminormedAddCommGroup V₄] [NormedSpace 𝕜 V₄] [PseudoMetricSpace P₄] [NormedAddTorsor V₄ P₄] (f : P₃ →ᵃⁱ[𝕜] P₄) (g : P₂ →ᵃⁱ[𝕜] P₃) (h : P →ᵃⁱ[𝕜] P₂)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- AffineIsometry.coe_one
@[simp]
theorem coe_one : ⇑(1 : P →ᵃⁱ[𝕜] P) = _root_.id 
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: ⇑1 = id
-- AffineIsometry.coe_mul
@[simp]
theorem coe_mul (f g : P →ᵃⁱ[𝕜] P) : ⇑(f * g) = f ∘ g 
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] (f g : P →ᵃⁱ[𝕜] P)
Goal: ⇑(f * g) = ⇑f ∘ ⇑g
-- AffineSubspace.subtypeₐᵢ_linear
theorem subtypeₐᵢ_linear (s : AffineSubspace 𝕜 P) [Nonempty s] :
    s.subtypeₐᵢ.linear = s.direction.subtype 
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] (s : AffineSubspace 𝕜 P) [Nonempty ↥s]
Goal: s.subtypeₐᵢ.linear = s.direction.subtype
-- AffineSubspace.subtypeₐᵢ_linearIsometry
@[simp]
theorem subtypeₐᵢ_linearIsometry (s : AffineSubspace 𝕜 P) [Nonempty s] :
    s.subtypeₐᵢ.linearIsometry = s.direction.subtypeₗᵢ 
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] (s : AffineSubspace 𝕜 P) [Nonempty ↥s]
Goal: s.subtypeₐᵢ.linearIsometry = s.direction.subtypeₗᵢ
-- AffineSubspace.coe_subtypeₐᵢ
@[simp]
theorem coe_subtypeₐᵢ (s : AffineSubspace 𝕜 P) [Nonempty s] : ⇑s.subtypeₐᵢ = s.subtype 
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] (s : AffineSubspace 𝕜 P) [Nonempty ↥s]
Goal: ⇑s.subtypeₐᵢ = ⇑s.subtype
-- AffineSubspace.subtypeₐᵢ_toAffineMap
@[simp]
theorem subtypeₐᵢ_toAffineMap (s : AffineSubspace 𝕜 P) [Nonempty s] :
    s.subtypeₐᵢ.toAffineMap = s.subtype 
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] (s : AffineSubspace 𝕜 P) [Nonempty ↥s]
Goal: s.subtypeₐᵢ.toAffineMap = s.subtype
-- AffineIsometryEquiv.norm_map
structure AffineIsometryEquiv extends P ≃ᵃ[𝕜] P₂ where
  norm_map : ∀ x, ‖linear x‖ = ‖x‖
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] (self : P ≃ᵃⁱ[𝕜] P₂) (x : V)
Goal: ‖self.linear x‖ = ‖x‖
-- AffineIsometryEquiv.linear_eq_linear_isometry
@[simp]
theorem linear_eq_linear_isometry : e.linear = e.linearIsometryEquiv.toLinearEquiv 
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: e.linear = e.linearIsometryEquiv.toLinearEquiv
-- AffineIsometryEquiv.coe_mk
@[simp]
theorem coe_mk (e : P ≃ᵃ[𝕜] P₂) (he : ∀ x, ‖e.linear x‖ = ‖x‖) : ⇑(mk e he) = 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 ≃ᵃ[𝕜] P₂) (he : ∀ (x : V), ‖e.linear x‖ = ‖x‖)
Goal: ⇑{ toAffineEquiv := e, norm_map := he } = ⇑e
-- AffineIsometryEquiv.coe_toAffineEquiv
@[simp]
theorem coe_toAffineEquiv (e : P ≃ᵃⁱ[𝕜] P₂) : ⇑e.toAffineEquiv = 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: ⇑e.toAffineEquiv = ⇑e
-- AffineIsometryEquiv.toAffineEquiv_injective
theorem toAffineEquiv_injective : Injective (toAffineEquiv : (P ≃ᵃⁱ[𝕜] P₂) → P ≃ᵃ[𝕜] P₂)
  
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]
Goal: Function.Injective AffineIsometryEquiv.toAffineEquiv
-- AffineIsometryEquiv.ext
@[ext]
theorem ext {e e' : P ≃ᵃⁱ[𝕜] P₂} (h : ∀ x, e x = e' x) : e = 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 e' : P ≃ᵃⁱ[𝕜] P₂} (h : ∀ (x : P), e x = e' x)
Goal: e = e'
-- AffineIsometryEquiv.coe_toAffineIsometry
@[simp]
theorem coe_toAffineIsometry : ⇑e.toAffineIsometry = 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: ⇑e.toAffineIsometry = ⇑e
-- AffineIsometryEquiv.coe_mk'
@[simp]
theorem coe_mk' (e : P₁ → P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p h) : ⇑(mk' e e' p h) = e 
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 : P₁ → P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p : P₁) (h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p)
Goal: ⇑(AffineIsometryEquiv.mk' e e' p h) = e
-- AffineIsometryEquiv.linearIsometryEquiv_mk'
@[simp]
theorem linearIsometryEquiv_mk' (e : P₁ → P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p h) :
    (mk' e e' p h).linearIsometryEquiv = e' 
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 : P₁ → P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p : P₁) (h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p)
Goal: (AffineIsometryEquiv.mk' e e' p h).linearIsometryEquiv = e'
-- LinearIsometryEquiv.coe_toAffineIsometryEquiv
@[simp]
theorem coe_toAffineIsometryEquiv : ⇑(e.toAffineIsometryEquiv : V ≃ᵃⁱ[𝕜] V₂) = e 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂)
Goal: ⇑e.toAffineIsometryEquiv = ⇑e
-- LinearIsometryEquiv.toAffineIsometryEquiv_linearIsometryEquiv
@[simp]
theorem toAffineIsometryEquiv_linearIsometryEquiv :
    e.toAffineIsometryEquiv.linearIsometryEquiv = e 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂)
Goal: e.toAffineIsometryEquiv.linearIsometryEquiv = e
-- LinearIsometryEquiv.toAffineIsometryEquiv_toAffineEquiv
@[simp]
theorem toAffineIsometryEquiv_toAffineEquiv :
    e.toAffineIsometryEquiv.toAffineEquiv = e.toLinearEquiv.toAffineEquiv 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂)
Goal: e.toAffineIsometryEquiv.toAffineEquiv = e.toAffineEquiv
-- LinearIsometryEquiv.toAffineIsometryEquiv_toAffineIsometry
@[simp]
theorem toAffineIsometryEquiv_toAffineIsometry :
    e.toAffineIsometryEquiv.toAffineIsometry = e.toLinearIsometry.toAffineIsometry 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂)
Goal: e.toAffineIsometryEquiv.toAffineIsometry = e.toLinearIsometry.toAffineIsometry
-- AffineIsometryEquiv.isometry
protected theorem isometry : Isometry 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: Isometry ⇑e
-- AffineIsometryEquiv.coe_toIsometryEquiv
@[simp]
theorem coe_toIsometryEquiv : ⇑e.toIsometryEquiv = 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: ⇑e.toIsometryEquiv = ⇑e
-- AffineIsometryEquiv.range_eq_univ
theorem range_eq_univ (e : P ≃ᵃⁱ[𝕜] P₂) : Set.range e = Set.univ 
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: Set.range ⇑e = Set.univ
-- AffineIsometryEquiv.coe_toHomeomorph
@[simp]
theorem coe_toHomeomorph : ⇑e.toHomeomorph = 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: ⇑e.toHomeomorph = ⇑e
-- AffineIsometryEquiv.continuous
protected theorem continuous : Continuous 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: Continuous ⇑e
-- AffineIsometryEquiv.continuousAt
protected theorem continuousAt {x} : ContinuousAt e x 
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 : P}
Goal: ContinuousAt (⇑e) x
-- AffineIsometryEquiv.continuousOn
protected theorem continuousOn {s} : ContinuousOn e 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: ContinuousOn (⇑e) s
-- AffineIsometryEquiv.continuousWithinAt
protected theorem continuousWithinAt {s x} : ContinuousWithinAt e s x 
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} {x : P}
Goal: ContinuousWithinAt (⇑e) s x
-- AffineIsometryEquiv.coe_refl
@[simp]
theorem coe_refl : ⇑(refl 𝕜 P) = id 
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.refl 𝕜 P) = id
-- AffineIsometryEquiv.toAffineEquiv_refl
@[simp]
theorem toAffineEquiv_refl : (refl 𝕜 P).toAffineEquiv = AffineEquiv.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.refl 𝕜 P).toAffineEquiv = AffineEquiv.refl 𝕜 P
-- AffineIsometryEquiv.toIsometryEquiv_refl
@[simp]
theorem toIsometryEquiv_refl : (refl 𝕜 P).toIsometryEquiv = IsometryEquiv.refl 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.refl 𝕜 P).toIsometryEquiv = IsometryEquiv.refl P
-- AffineIsometryEquiv.toHomeomorph_refl
@[simp]
theorem toHomeomorph_refl : (refl 𝕜 P).toHomeomorph = Homeomorph.refl 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.refl 𝕜 P).toHomeomorph = Homeomorph.refl P
-- AffineIsometryEquiv.apply_symm_apply
@[simp]
theorem apply_symm_apply (x : P₂) : e (e.symm x) = x 
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 : P₂)
Goal: e (e.symm x) = x
-- AffineIsometryEquiv.symm_apply_apply
@[simp]
theorem symm_apply_apply (x : P) : e.symm (e x) = x 
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 : P)
Goal: e.symm (e x) = x
-- AffineIsometryEquiv.symm_symm
@[simp]
theorem symm_symm : e.symm.symm = 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: e.symm.symm = e
-- AffineIsometryEquiv.toAffineEquiv_symm
@[simp]
theorem toAffineEquiv_symm : e.toAffineEquiv.symm = e.symm.toAffineEquiv 
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: e.symm = e.symm.toAffineEquiv
-- AffineIsometryEquiv.toIsometryEquiv_symm
@[simp]
theorem toIsometryEquiv_symm : e.toIsometryEquiv.symm = e.symm.toIsometryEquiv 
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: e.toIsometryEquiv.symm = e.symm.toIsometryEquiv
-- AffineIsometryEquiv.toHomeomorph_symm
@[simp]
theorem toHomeomorph_symm : e.toHomeomorph.symm = e.symm.toHomeomorph 
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: e.toHomeomorph.symm = e.symm.toHomeomorph
-- AffineIsometryEquiv.coe_trans
@[simp]
theorem coe_trans (e₁ : P ≃ᵃⁱ[𝕜] P₂) (e₂ : P₂ ≃ᵃⁱ[𝕜] P₃) : ⇑(e₁.trans e₂) = e₂ ∘ e₁ 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P₃] [NormedAddTorsor V₃ P₃] (e₁ : P ≃ᵃⁱ[𝕜] P₂) (e₂ : P₂ ≃ᵃⁱ[𝕜] P₃)
Goal: ⇑(e₁.trans e₂) = ⇑e₂ ∘ ⇑e₁
-- AffineIsometryEquiv.trans_refl
@[simp]
theorem trans_refl : e.trans (refl 𝕜 P₂) = 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: e.trans (AffineIsometryEquiv.refl 𝕜 P₂) = e
-- AffineIsometryEquiv.refl_trans
@[simp]
theorem refl_trans : (refl 𝕜 P).trans e = 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: (AffineIsometryEquiv.refl 𝕜 P).trans e = e
-- AffineIsometryEquiv.self_trans_symm
@[simp]
theorem self_trans_symm : e.trans e.symm = refl 𝕜 P 
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: e.trans e.symm = AffineIsometryEquiv.refl 𝕜 P
-- AffineIsometryEquiv.symm_trans_self
@[simp]
theorem symm_trans_self : e.symm.trans e = refl 𝕜 P₂ 
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: e.symm.trans e = AffineIsometryEquiv.refl 𝕜 P₂
-- AffineIsometryEquiv.coe_symm_trans
@[simp]
theorem coe_symm_trans (e₁ : P ≃ᵃⁱ[𝕜] P₂) (e₂ : P₂ ≃ᵃⁱ[𝕜] P₃) :
    ⇑(e₁.trans e₂).symm = e₁.symm ∘ e₂.symm 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P₃] [NormedAddTorsor V₃ P₃] (e₁ : P ≃ᵃⁱ[𝕜] P₂) (e₂ : P₂ ≃ᵃⁱ[𝕜] P₃)
Goal: ⇑(e₁.trans e₂).symm = ⇑e₁.symm ∘ ⇑e₂.symm
-- AffineIsometryEquiv.trans_assoc
theorem trans_assoc (ePP₂ : P ≃ᵃⁱ[𝕜] P₂) (eP₂G : P₂ ≃ᵃⁱ[𝕜] P₃) (eGG' : P₃ ≃ᵃⁱ[𝕜] P₄) :
    ePP₂.trans (eP₂G.trans eGG') = (ePP₂.trans eP₂G).trans eGG' 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {V₄ : Type u_7} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} {P₄ : Type u_13} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P₃] [NormedAddTorsor V₃ P₃] [SeminormedAddCommGroup V₄] [NormedSpace 𝕜 V₄] [PseudoMetricSpace P₄] [NormedAddTorsor V₄ P₄] (ePP₂ : P ≃ᵃⁱ[𝕜] P₂) (eP₂G : P₂ ≃ᵃⁱ[𝕜] P₃) (eGG' : P₃ ≃ᵃⁱ[𝕜] P₄)
Goal: ePP₂.trans (eP₂G.trans eGG') = (ePP₂.trans eP₂G).trans eGG'
-- AffineIsometryEquiv.coe_one
@[simp]
theorem coe_one : ⇑(1 : P ≃ᵃⁱ[𝕜] P) = id 
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: ⇑1 = id
-- AffineIsometryEquiv.coe_mul
@[simp]
theorem coe_mul (e e' : P ≃ᵃⁱ[𝕜] P) : ⇑(e * e') = e ∘ e' 
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] (e e' : P ≃ᵃⁱ[𝕜] P)
Goal: ⇑(e * e') = ⇑e ∘ ⇑e'
-- AffineIsometryEquiv.coe_inv
@[simp]
theorem coe_inv (e : P ≃ᵃⁱ[𝕜] P) : ⇑e⁻¹ = e.symm 
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] (e : P ≃ᵃⁱ[𝕜] P)
Goal: ⇑e⁻¹ = ⇑e.symm
-- AffineIsometryEquiv.map_vadd
@[simp]
theorem map_vadd (p : P) (v : V) : e (v +ᵥ p) = e.linearIsometryEquiv v +ᵥ e p 
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) (p : P) (v : V)
Goal: e (v +ᵥ p) = e.linearIsometryEquiv v +ᵥ e p
-- AffineIsometryEquiv.map_vsub
@[simp]
theorem map_vsub (p1 p2 : P) : e.linearIsometryEquiv (p1 -ᵥ p2) = e p1 -ᵥ e p2 
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) (p1 p2 : P)
Goal: e.linearIsometryEquiv (p1 -ᵥ p2) = e p1 -ᵥ e p2
-- AffineIsometryEquiv.dist_map
@[simp]
theorem dist_map (x y : P) : dist (e x) (e y) = dist 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: dist (e x) (e y) = dist x y
-- AffineIsometryEquiv.edist_map
@[simp]
theorem edist_map (x y : P) : edist (e x) (e y) = edist 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: edist (e x) (e y) = edist x y