Theorem: EuclideanGeometry.mul_dist_add_mul_dist_eq_mul_dist_of_cospherical
File: Mathlib/Geometry/Euclidean/Sphere/Ptolemy.lean

Proof steps:
state_before: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
⊢ dist a b * dist c d + dist b c * dist d a = dist a c * dist b d
state_after: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
⊢ dist a b * dist c d + dist b c * dist d a = dist a c * dist b d
tactic: have h' : Cospherical ({a, c, b, d} : Set P) := by rwa [Set.insert_comm c b {d}]
state_before: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
⊢ dist a b * dist c d + dist b c * dist d a = dist a c * dist b d
state_after: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
⊢ dist a b * dist c d + dist b c * dist d a = dist a c * dist b d
tactic: have hmul := mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi h' hapc hbpd
state_before: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
⊢ dist a b * dist c d + dist b c * dist d a = dist a c * dist b d
state_after: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
⊢ dist a b * dist c d + dist b c * dist d a = dist a c * dist b d
tactic: have hbp := left_dist_ne_zero_of_angle_eq_pi hbpd
state_before: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
h₁ : dist c d = dist c p / dist b p * dist a b
h₂ : dist d a = dist a p / dist b p * dist b c
⊢ dist a b * dist c d + dist b c * dist d a = dist a c * dist b d
state_after: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
h₁ : dist c d = dist c p / dist b p * dist a b
h₂ : dist d a = dist a p / dist b p * dist b c
h₃ : dist d p = dist a p * dist c p / dist b p
⊢ dist a b * dist c d + dist b c * dist d a = dist a c * dist b d
tactic: have h₃ : dist d p = dist a p * dist c p / dist b p := by field_simp [mul_comm, hmul]
state_before: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
h₁ : dist c d = dist c p / dist b p * dist a b
h₂ : dist d a = dist a p / dist b p * dist b c
h₃ : dist d p = dist a p * dist c p / dist b p
⊢ dist a b * dist c d + dist b c * dist d a = dist a c * dist b d
state_after: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
h₁ : dist c d = dist c p / dist b p * dist a b
h₂ : dist d a = dist a p / dist b p * dist b c
h₃ : dist d p = dist a p * dist c p / dist b p
h₄ : ∀ (x y : ℝ), x * (y * x) = x * x * y
⊢ dist a b * dist c d + dist b c * dist d a = dist a c * dist b d
tactic: have h₄ : ∀ x y : ℝ, x * (y * x) = x * x * y := fun x y => by rw [mul_left_comm, mul_comm]
state_before: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapc : ∠ a p c = π
hbpd : ∠ b p d = π
h' : Cospherical {a, c, b, d}
hmul : dist a p * dist c p = dist b p * dist d p
hbp : dist b p ≠ 0
h₁ : dist c d = dist c p / dist b p * dist a b
h₂ : dist d a = dist a p / dist b p * dist b c
h₃ : dist d p = dist a p * dist c p / dist b p
h₄ : ∀ (x y : ℝ), x * (y * x) = x * x * y
⊢ dist a b * dist c d + dist b c * dist d a = dist a c * dist b d
state_after: no goals
tactic: field_simp [h₁, h₂, dist_eq_add_dist_of_angle_eq_pi hbpd, h₃, hbp, dist_comm a b, h₄, ← sq,
    dist_sq_mul_dist_add_dist_sq_mul_dist b, hapc]