[GOAL]
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
⊢ ∀ (e : PE ≃ᵢ PE), ↑e x = x → ↑e y = y → ↑e (midpoint ℝ x y) = midpoint ℝ x y
[PROOFSTEP]
set z := midpoint ℝ x y
[GOAL]
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
⊢ ∀ (e : PE ≃ᵢ PE), ↑e x = x → ↑e y = y → ↑e z = z
[PROOFSTEP]
set s := {e : PE ≃ᵢ PE | e x = x ∧ e y = y}
[GOAL]
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}
⊢ ∀ (e : PE ≃ᵢ PE), ↑e x = x → ↑e y = y → ↑e z = z
[PROOFSTEP]
haveI : Nonempty s :=
  ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
    -- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
[GOAL]
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
[PROOFSTEP]
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]`.
[GOAL]
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)
[PROOFSTEP]
refine' ⟨dist x z + dist x z, forall_range_iff.2 <| Subtype.forall.2 _⟩
[GOAL]
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
[PROOFSTEP]
rintro e ⟨hx, _⟩
[GOAL]
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
[PROOFSTEP]
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]`.
[GOAL]
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
[PROOFSTEP]
rw [hx, dist_comm]
[GOAL]
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
[PROOFSTEP]
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]`.
[GOAL]
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
[PROOFSTEP]
set R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
[GOAL]
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
[PROOFSTEP]
set f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e => ((e.trans R).trans e.symm).trans R
[GOAL]
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
[PROOFSTEP]
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
[GOAL]
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
[PROOFSTEP]
intro e
[GOAL]
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
[PROOFSTEP]
dsimp
[GOAL]
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)
[PROOFSTEP]
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
[GOAL]
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
[PROOFSTEP]
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]
      -- Therefore, `dist (e z) z = 0` for all `e ∈ s`.
[GOAL]
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
⊢ MapsTo f s s
[PROOFSTEP]
rintro e ⟨hx, hy⟩
[GOAL]
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
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
hx : ↑e x = x
hy : ↑e y = y
⊢ f e ∈ s
[PROOFSTEP]
constructor
[GOAL]
case intro.left
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
hx : ↑e x = x
hy : ↑e y = y
⊢ ↑(f e) x = x
[PROOFSTEP]
simp [hx, hy, e.symm_apply_eq.2 hx.symm, e.symm_apply_eq.2 hy.symm]
  -- Therefore, `dist (e z) z = 0` for all `e ∈ s`.
[GOAL]
case intro.right
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
hx : ↑e x = x
hy : ↑e y = y
⊢ ↑(f e) y = y
[PROOFSTEP]
simp [hx, hy, e.symm_apply_eq.2 hx.symm, e.symm_apply_eq.2 hy.symm]
  -- Therefore, `dist (e z) z = 0` for all `e ∈ s`.
[GOAL]
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
hf_maps_to : MapsTo f s s
⊢ ∀ (e : PE ≃ᵢ PE), ↑e x = x → ↑e y = y → ↑e z = z
[PROOFSTEP]
set c := ⨆ e : s, dist ((e : PE ≃ᵢ PE) z) z
[GOAL]
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
hf_maps_to : MapsTo f s s
c : ℝ := ⨆ (e : ↑s), dist (↑↑e z) z
⊢ ∀ (e : PE ≃ᵢ PE), ↑e x = x → ↑e y = y → ↑e z = z
[PROOFSTEP]
have : c ≤ c / 2 := by
  apply ciSup_le
  rintro ⟨e, he⟩
  simp only [Subtype.coe_mk, le_div_iff' (zero_lt_two' ℝ), ← hf_dist]
  exact le_ciSup h_bdd ⟨f e, hf_maps_to he⟩
[GOAL]
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
hf_maps_to : MapsTo f s s
c : ℝ := ⨆ (e : ↑s), dist (↑↑e z) z
⊢ c ≤ c / 2
[PROOFSTEP]
apply ciSup_le
[GOAL]
case H
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
hf_maps_to : MapsTo f s s
c : ℝ := ⨆ (e : ↑s), dist (↑↑e z) z
⊢ ∀ (x : ↑s), dist (↑↑x z) z ≤ c / 2
[PROOFSTEP]
rintro ⟨e, he⟩
[GOAL]
case H.mk
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
hf_maps_to : MapsTo f s s
c : ℝ := ⨆ (e : ↑s), dist (↑↑e z) z
e : PE ≃ᵢ PE
he : e ∈ s
⊢ dist (↑↑{ val := e, property := he } z) z ≤ c / 2
[PROOFSTEP]
simp only [Subtype.coe_mk, le_div_iff' (zero_lt_two' ℝ), ← hf_dist]
[GOAL]
case H.mk
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
hf_maps_to : MapsTo f s s
c : ℝ := ⨆ (e : ↑s), dist (↑↑e z) z
e : PE ≃ᵢ PE
he : e ∈ s
⊢ dist
      (↑(IsometryEquiv.trans
            (IsometryEquiv.trans (IsometryEquiv.trans e (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y))))
              (IsometryEquiv.symm e))
            (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y))))
        (midpoint ℝ x y))
      (midpoint ℝ x y) ≤
    ⨆ (e : ↑{e | ↑e x = x ∧ ↑e y = y}), dist (↑↑e (midpoint ℝ x y)) (midpoint ℝ x y)
[PROOFSTEP]
exact le_ciSup h_bdd ⟨f e, hf_maps_to he⟩
[GOAL]
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
hf_maps_to : MapsTo f s s
c : ℝ := ⨆ (e : ↑s), dist (↑↑e z) z
this : c ≤ c / 2
⊢ ∀ (e : PE ≃ᵢ PE), ↑e x = x → ↑e y = y → ↑e z = z
[PROOFSTEP]
replace : c ≤ 0
[GOAL]
case this
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
hf_maps_to : MapsTo f s s
c : ℝ := ⨆ (e : ↑s), dist (↑↑e z) z
this : c ≤ c / 2
⊢ c ≤ 0
[PROOFSTEP]
linarith
[GOAL]
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
hf_maps_to : MapsTo f s s
c : ℝ := ⨆ (e : ↑s), dist (↑↑e z) z
this : c ≤ 0
⊢ ∀ (e : PE ≃ᵢ PE), ↑e x = x → ↑e y = y → ↑e z = z
[PROOFSTEP]
refine' fun e hx hy => dist_le_zero.1 (le_trans _ this)
[GOAL]
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
hf_maps_to : MapsTo f s s
c : ℝ := ⨆ (e : ↑s), dist (↑↑e z) z
this : c ≤ 0
e : PE ≃ᵢ PE
hx : ↑e x = x
hy : ↑e y = y
⊢ dist (↑e z) z ≤ c
[PROOFSTEP]
exact le_ciSup h_bdd ⟨e, hx, hy⟩
[GOAL]
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
f : PE ≃ᵢ PF
x y : PE
⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)
[PROOFSTEP]
set e : PE ≃ᵢ PE :=
  ((f.trans <| (pointReflection ℝ <| midpoint ℝ (f x) (f y)).toIsometryEquiv).trans f.symm).trans
    (pointReflection ℝ <| midpoint ℝ x y).toIsometryEquiv
[GOAL]
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
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  IsometryEquiv.trans
    (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))
      (IsometryEquiv.symm f))
    (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))
⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)
[PROOFSTEP]
have hx : e x = x := by simp
[GOAL]
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
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  IsometryEquiv.trans
    (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))
      (IsometryEquiv.symm f))
    (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))
⊢ ↑e x = x
[PROOFSTEP]
simp
[GOAL]
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
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  IsometryEquiv.trans
    (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))
      (IsometryEquiv.symm f))
    (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))
hx : ↑e x = x
⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)
[PROOFSTEP]
have hy : e y = y := by simp
[GOAL]
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
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  IsometryEquiv.trans
    (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))
      (IsometryEquiv.symm f))
    (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))
hx : ↑e x = x
⊢ ↑e y = y
[PROOFSTEP]
simp
[GOAL]
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
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  IsometryEquiv.trans
    (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))
      (IsometryEquiv.symm f))
    (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))
hx : ↑e x = x
hy : ↑e y = y
⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)
[PROOFSTEP]
have hm := e.midpoint_fixed hx hy
[GOAL]
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
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  IsometryEquiv.trans
    (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))
      (IsometryEquiv.symm f))
    (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))
hx : ↑e x = x
hy : ↑e y = y
hm : ↑e (midpoint ℝ x y) = midpoint ℝ x y
⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)
[PROOFSTEP]
simp only [trans_apply] at hm 
[GOAL]
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
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  IsometryEquiv.trans
    (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))
      (IsometryEquiv.symm f))
    (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))
hx : ↑e x = x
hy : ↑e y = y
hm :
  ↑(toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))
      (↑(IsometryEquiv.symm f)
        (↑(toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))) (↑f (midpoint ℝ x y)))) =
    midpoint ℝ x y
⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)
[PROOFSTEP]
rwa [← eq_symm_apply, toIsometryEquiv_symm, pointReflection_symm, coe_toIsometryEquiv, coe_toIsometryEquiv,
  pointReflection_self, symm_apply_eq, @pointReflection_fixed_iff] at hm 
[GOAL]
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
f : E ≃ᵢ F
h0 : ↑f 0 = 0
src✝ : E →L[ℝ] F :=
  AddMonoidHom.toRealLinearMap
    (AddMonoidHom.ofMapMidpoint ℝ ℝ (↑f) h0 (_ : ∀ (x y : E), ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)))
    (_ : Continuous ↑f)
x : E
⊢ ‖↑f x‖ = ‖x‖
[PROOFSTEP]
simp only [← dist_zero_right, ← h0, f.dist_eq]
[GOAL]
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
f : E ≃ᵢ F
⊢ ↑(IsometryEquiv.trans f (IsometryEquiv.symm (addRight (↑f 0)))) 0 = 0
[PROOFSTEP]
simpa only [sub_eq_add_neg] using sub_self (f 0)
[GOAL]
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
f : PE ≃ᵢ PF
p : PE
⊢ ↑f p =
    ↑(toRealLinearIsometryEquiv
            (IsometryEquiv.trans (vaddConst (Classical.arbitrary PE))
              (IsometryEquiv.trans f (IsometryEquiv.symm (vaddConst (↑f (Classical.arbitrary PE)))))))
        (p -ᵥ Classical.arbitrary PE) +ᵥ
      ↑f (Classical.arbitrary PE)
[PROOFSTEP]
simp
[GOAL]
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
f : PE ≃ᵢ PF
⊢ toIsometryEquiv (toRealAffineIsometryEquiv f) = f
[PROOFSTEP]
ext
[GOAL]
case H
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
f : PE ≃ᵢ PF
x✝ : PE
⊢ ↑(toIsometryEquiv (toRealAffineIsometryEquiv f)) x✝ = ↑f x✝
[PROOFSTEP]
rfl
