l3lab/ntp-mathlib huggingface dataset:
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  

Proof state:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e z = z

Next tactic:
have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
    rintro e ⟨hx, _⟩
    calc
      dist (e z) z ≤ dist (e z) x + dist x z := dist_triangle (e z) x z
      _ = dist (e x) (e z) + dist x z := by rw [hx, dist_comm]
      _ = dist x z + dist x z := by erw [e.dist_eq x z]
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    

Proof state:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
⊢ BddAbove (range fun e => dist (↑e z) z)

Next tactic:
refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
    

Proof state:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
⊢ ∀ (a : PE ≃ᵢ PE) (b : a ∈ s), dist (↑{ val := a, property := b } z) z ≤ dist x z + dist x z

Next tactic:
rintro e ⟨hx, _⟩
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
    rintro e ⟨hx, _⟩
    

Proof state:
case intro
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
e : PE ≃ᵢ PE
hx : e x = x
right✝ : e y = y
⊢ dist (↑{ val := e, property := (_ : e x = x ∧ e y = y) } z) z ≤ dist x z + dist x z

Next tactic:
calc
      dist (e z) z ≤ dist (e z) x + dist x z := dist_triangle (e z) x z
      _ = dist (e x) (e z) + dist x z := by rw [hx, dist_comm]
      _ = dist x z + dist x z := by erw [e.dist_eq x z]
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
    rintro e ⟨hx, _⟩
    calc
      dist (e z) z ≤ dist (e z) x + dist x z := dist_triangle (e z) x z
      _ = dist (e x) (e z) + dist x z := by 

Proof state:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
e : PE ≃ᵢ PE
hx : e x = x
right✝ : e y = y
⊢ dist (e z) x + dist x z = dist (e x) (e z) + dist x z

Next tactic:
rw [hx, dist_comm]
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
    rintro e ⟨hx, _⟩
    calc
      dist (e z) z ≤ dist (e z) x + dist x z := dist_triangle (e z) x z
      _ = dist (e x) (e z) + dist x z := by rw [hx, dist_comm]
      _ = dist x z + dist x z := by 

Proof state:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
e : PE ≃ᵢ PE
hx : e x = x
right✝ : e y = y
⊢ dist (e x) (e z) + dist x z = dist x z + dist x z

Next tactic:
erw [e.dist_eq x z]
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
    rintro e ⟨hx, _⟩
    calc
      dist (e z) z ≤ dist (e z) x + dist x z := dist_triangle (e z) x z
      _ = dist (e x) (e z) + dist x z := by rw [hx, dist_comm]
      _ = dist x z + dist x z := by erw [e.dist_eq x z]
  -- On the other hand, consider the map `f : (E ≃ᵢ E) → (E ≃ᵢ E)`
  -- sending each `e` to `R ∘ e⁻¹ ∘ R ∘ e`, where `R` is the point reflection in the
  -- midpoint `z` of `[x, y]`.
  

Proof state:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e => dist (↑e z) z)
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e z = z

Next tactic:
set R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
    rintro e ⟨hx, _⟩
    calc
      dist (e z) z ≤ dist (e z) x + dist x z := dist_triangle (e z) x z
      _ = dist (e x) (e z) + dist x z := by rw [hx, dist_comm]
      _ = dist x z + dist x z := by erw [e.dist_eq x z]
  -- On the other hand, consider the map `f : (E ≃ᵢ E) → (E ≃ᵢ E)`
  -- sending each `e` to `R ∘ e⁻¹ ∘ R ∘ e`, where `R` is the point reflection in the
  -- midpoint `z` of `[x, y]`.
  set R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
  

Proof state:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e => dist (↑e z) z)
R : PE ≃ᵢ PE := toIsometryEquiv (pointReflection ℝ z)
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e z = z

Next tactic:
set f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e => ((e.trans R).trans e.symm).trans R
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
    rintro e ⟨hx, _⟩
    calc
      dist (e z) z ≤ dist (e z) x + dist x z := dist_triangle (e z) x z
      _ = dist (e x) (e z) + dist x z := by rw [hx, dist_comm]
      _ = dist x z + dist x z := by erw [e.dist_eq x z]
  -- On the other hand, consider the map `f : (E ≃ᵢ E) → (E ≃ᵢ E)`
  -- sending each `e` to `R ∘ e⁻¹ ∘ R ∘ e`, where `R` is the point reflection in the
  -- midpoint `z` of `[x, y]`.
  set R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
  set f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e => ((e.trans R).trans e.symm).trans R
  -- Note that `f` doubles the value of `dist (e z) z`
  

Proof state:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e => dist (↑e z) z)
R : PE ≃ᵢ PE := toIsometryEquiv (pointReflection ℝ z)
f : PE ≃ᵢ PE → PE ≃ᵢ PE :=
  fun e => IsometryEquiv.trans (IsometryEquiv.trans (IsometryEquiv.trans e R) (IsometryEquiv.symm e)) R
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e z = z

Next tactic:
have hf_dist : ∀ e, dist (f e z) z = 2 * dist (e z) z := by
    intro e
    dsimp
    rw [dist_pointReflection_fixed, ← e.dist_eq, e.apply_symm_apply,
      dist_pointReflection_self_real, dist_comm]
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
    rintro e ⟨hx, _⟩
    calc
      dist (e z) z ≤ dist (e z) x + dist x z := dist_triangle (e z) x z
      _ = dist (e x) (e z) + dist x z := by rw [hx, dist_comm]
      _ = dist x z + dist x z := by erw [e.dist_eq x z]
  -- On the other hand, consider the map `f : (E ≃ᵢ E) → (E ≃ᵢ E)`
  -- sending each `e` to `R ∘ e⁻¹ ∘ R ∘ e`, where `R` is the point reflection in the
  -- midpoint `z` of `[x, y]`.
  set R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
  set f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e => ((e.trans R).trans e.symm).trans R
  -- Note that `f` doubles the value of `dist (e z) z`
  have hf_dist : ∀ e, dist (f e z) z = 2 * dist (e z) z := by
    

Proof state:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e => dist (↑e z) z)
R : PE ≃ᵢ PE := toIsometryEquiv (pointReflection ℝ z)
f : PE ≃ᵢ PE → PE ≃ᵢ PE :=
  fun e => IsometryEquiv.trans (IsometryEquiv.trans (IsometryEquiv.trans e R) (IsometryEquiv.symm e)) R
⊢ ∀ (e : PE ≃ᵢ PE), dist ((f e) z) z = 2 * dist (e z) z

Next tactic:
intro e
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
    rintro e ⟨hx, _⟩
    calc
      dist (e z) z ≤ dist (e z) x + dist x z := dist_triangle (e z) x z
      _ = dist (e x) (e z) + dist x z := by rw [hx, dist_comm]
      _ = dist x z + dist x z := by erw [e.dist_eq x z]
  -- On the other hand, consider the map `f : (E ≃ᵢ E) → (E ≃ᵢ E)`
  -- sending each `e` to `R ∘ e⁻¹ ∘ R ∘ e`, where `R` is the point reflection in the
  -- midpoint `z` of `[x, y]`.
  set R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
  set f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e => ((e.trans R).trans e.symm).trans R
  -- Note that `f` doubles the value of `dist (e z) z`
  have hf_dist : ∀ e, dist (f e z) z = 2 * dist (e z) z := by
    intro e
    

Proof state:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e => dist (↑e z) z)
R : PE ≃ᵢ PE := toIsometryEquiv (pointReflection ℝ z)
f : PE ≃ᵢ PE → PE ≃ᵢ PE :=
  fun e => IsometryEquiv.trans (IsometryEquiv.trans (IsometryEquiv.trans e R) (IsometryEquiv.symm e)) R
e : PE ≃ᵢ PE
⊢ dist ((f e) z) z = 2 * dist (e z) z

Next tactic:
dsimp
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
    rintro e ⟨hx, _⟩
    calc
      dist (e z) z ≤ dist (e z) x + dist x z := dist_triangle (e z) x z
      _ = dist (e x) (e z) + dist x z := by rw [hx, dist_comm]
      _ = dist x z + dist x z := by erw [e.dist_eq x z]
  -- On the other hand, consider the map `f : (E ≃ᵢ E) → (E ≃ᵢ E)`
  -- sending each `e` to `R ∘ e⁻¹ ∘ R ∘ e`, where `R` is the point reflection in the
  -- midpoint `z` of `[x, y]`.
  set R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
  set f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e => ((e.trans R).trans e.symm).trans R
  -- Note that `f` doubles the value of `dist (e z) z`
  have hf_dist : ∀ e, dist (f e z) z = 2 * dist (e z) z := by
    intro e
    dsimp
    

Proof state:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e => dist (↑e z) z)
R : PE ≃ᵢ PE := toIsometryEquiv (pointReflection ℝ z)
f : PE ≃ᵢ PE → PE ≃ᵢ PE :=
  fun e => IsometryEquiv.trans (IsometryEquiv.trans (IsometryEquiv.trans e R) (IsometryEquiv.symm e)) R
e : PE ≃ᵢ PE
⊢ dist
      ((pointReflection ℝ (midpoint ℝ x y))
        ((IsometryEquiv.symm e) ((pointReflection ℝ (midpoint ℝ x y)) (e (midpoint ℝ x y)))))
      (midpoint ℝ x y) =
    2 * dist (e (midpoint ℝ x y)) (midpoint ℝ x y)

Next tactic:
rw [dist_pointReflection_fixed, ← e.dist_eq, e.apply_symm_apply,
      dist_pointReflection_self_real, dist_comm]
/-- If an isometric self-homeomorphism of a normed vector space over `ℝ` fixes `x` and `y`,
then it fixes the midpoint of `[x, y]`. This is a lemma for a more general Mazur-Ulam theorem,
see below. -/
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by
  set z := midpoint ℝ x y
  -- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
  set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
  haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
  -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
  have h_bdd : BddAbove (range fun e : s => dist ((e : PE ≃ᵢ PE) z) z) := by
    refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
    rintro e ⟨hx, _⟩
    calc
      dist (e z) z ≤ dist (e z) x + dist x z := dist_triangle (e z) x z
      _ = dist (e x) (e z) + dist x z := by rw [hx, dist_comm]
      _ = dist x z + dist x z := by erw [e.dist_eq x z]
  -- On the other hand, consider the map `f : (E ≃ᵢ E) → (E ≃ᵢ E)`
  -- sending each `e` to `R ∘ e⁻¹ ∘ R ∘ e`, where `R` is the point reflection in the
  -- midpoint `z` of `[x, y]`.
  set R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
  set f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e => ((e.trans R).trans e.symm).trans R
  -- Note that `f` doubles the value of `dist (e z) z`
  have hf_dist : ∀ e, dist (f e z) z = 2 * dist (e z) z := by
    intro e
    dsimp
    rw [dist_pointReflection_fixed, ← e.dist_eq, e.apply_symm_apply,
      dist_pointReflection_self_real, dist_comm]
  -- Also note that `f` maps `s` to itself
  

Proof state:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e => dist (↑e z) z)
R : PE ≃ᵢ PE := toIsometryEquiv (pointReflection ℝ z)
f : PE ≃ᵢ PE → PE ≃ᵢ PE :=
  fun e => IsometryEquiv.trans (IsometryEquiv.trans (IsometryEquiv.trans e R) (IsometryEquiv.symm e)) R
hf_dist : ∀ (e : PE ≃ᵢ PE), dist ((f e) z) z = 2 * dist (e z) z
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e z = z

Next tactic:
have hf_maps_to : MapsTo f s s := by
    rintro e ⟨hx, hy⟩
    constructor <;> simp [hx, hy, e.symm_apply_eq.2 hx.symm, e.symm_apply_eq.2 hy.symm]