l3lab/ntp-mathlib huggingface dataset:
/-- The oriented angle between two vectors is zero or `π` if and only if the first vector is zero
or the second is a multiple of the first. -/
theorem oangle_eq_zero_or_eq_pi_iff_right_eq_smul {x y : V} :
    o.oangle x y = 0 ∨ o.oangle x y = π ↔ x = 0 ∨ ∃ r : ℝ, y = r • x := by
  rw [oangle_eq_zero_iff_sameRay, oangle_eq_pi_iff_sameRay_neg]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with (h | ⟨-, -, h⟩)
    · by_cases hx : x = 0; · simp [hx]
      obtain ⟨r, -, rfl⟩ := h.exists_nonneg_left hx
      exact Or.inr ⟨r, rfl⟩
    · by_cases hx : x = 0; · simp [hx]
      obtain ⟨r, -, hy⟩ := h.exists_nonneg_left hx
      refine' Or.inr ⟨-r, _⟩
      simp [hy]
  · rcases h with (rfl | ⟨r, rfl⟩); · simp
    by_cases hx : x = 0; · simp [hx]
    rcases lt_trichotomy r 0 with (hr | hr | hr)
    · 

Proof state:
case neg.inl
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x : V
r : ℝ
hx : ¬x = 0
hr : r < 0
⊢ SameRay ℝ x (r • x) ∨ x ≠ 0 ∧ r • x ≠ 0 ∧ SameRay ℝ x (-(r • x))

Next tactic:
rw [← neg_smul]
/-- The oriented angle between two vectors is zero or `π` if and only if the first vector is zero
or the second is a multiple of the first. -/
theorem oangle_eq_zero_or_eq_pi_iff_right_eq_smul {x y : V} :
    o.oangle x y = 0 ∨ o.oangle x y = π ↔ x = 0 ∨ ∃ r : ℝ, y = r • x := by
  rw [oangle_eq_zero_iff_sameRay, oangle_eq_pi_iff_sameRay_neg]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with (h | ⟨-, -, h⟩)
    · by_cases hx : x = 0; · simp [hx]
      obtain ⟨r, -, rfl⟩ := h.exists_nonneg_left hx
      exact Or.inr ⟨r, rfl⟩
    · by_cases hx : x = 0; · simp [hx]
      obtain ⟨r, -, hy⟩ := h.exists_nonneg_left hx
      refine' Or.inr ⟨-r, _⟩
      simp [hy]
  · rcases h with (rfl | ⟨r, rfl⟩); · simp
    by_cases hx : x = 0; · simp [hx]
    rcases lt_trichotomy r 0 with (hr | hr | hr)
    · rw [← neg_smul]
      

Proof state:
case neg.inl
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x : V
r : ℝ
hx : ¬x = 0
hr : r < 0
⊢ SameRay ℝ x (r • x) ∨ x ≠ 0 ∧ r • x ≠ 0 ∧ SameRay ℝ x (-r • x)

Next tactic:
exact Or.inr ⟨hx, smul_ne_zero hr.ne hx,
        SameRay.sameRay_pos_smul_right x (Left.neg_pos_iff.2 hr)⟩
/-- The oriented angle between two vectors is zero or `π` if and only if the first vector is zero
or the second is a multiple of the first. -/
theorem oangle_eq_zero_or_eq_pi_iff_right_eq_smul {x y : V} :
    o.oangle x y = 0 ∨ o.oangle x y = π ↔ x = 0 ∨ ∃ r : ℝ, y = r • x := by
  rw [oangle_eq_zero_iff_sameRay, oangle_eq_pi_iff_sameRay_neg]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with (h | ⟨-, -, h⟩)
    · by_cases hx : x = 0; · simp [hx]
      obtain ⟨r, -, rfl⟩ := h.exists_nonneg_left hx
      exact Or.inr ⟨r, rfl⟩
    · by_cases hx : x = 0; · simp [hx]
      obtain ⟨r, -, hy⟩ := h.exists_nonneg_left hx
      refine' Or.inr ⟨-r, _⟩
      simp [hy]
  · rcases h with (rfl | ⟨r, rfl⟩); · simp
    by_cases hx : x = 0; · simp [hx]
    rcases lt_trichotomy r 0 with (hr | hr | hr)
    · rw [← neg_smul]
      exact Or.inr ⟨hx, smul_ne_zero hr.ne hx,
        SameRay.sameRay_pos_smul_right x (Left.neg_pos_iff.2 hr)⟩
    · 

Proof state:
case neg.inr.inl
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x : V
r : ℝ
hx : ¬x = 0
hr : r = 0
⊢ SameRay ℝ x (r • x) ∨ x ≠ 0 ∧ r • x ≠ 0 ∧ SameRay ℝ x (-(r • x))

Next tactic:
simp [hr]
/-- The oriented angle between two vectors is zero or `π` if and only if the first vector is zero
or the second is a multiple of the first. -/
theorem oangle_eq_zero_or_eq_pi_iff_right_eq_smul {x y : V} :
    o.oangle x y = 0 ∨ o.oangle x y = π ↔ x = 0 ∨ ∃ r : ℝ, y = r • x := by
  rw [oangle_eq_zero_iff_sameRay, oangle_eq_pi_iff_sameRay_neg]
  refine' ⟨fun h => _, fun h => _⟩
  · rcases h with (h | ⟨-, -, h⟩)
    · by_cases hx : x = 0; · simp [hx]
      obtain ⟨r, -, rfl⟩ := h.exists_nonneg_left hx
      exact Or.inr ⟨r, rfl⟩
    · by_cases hx : x = 0; · simp [hx]
      obtain ⟨r, -, hy⟩ := h.exists_nonneg_left hx
      refine' Or.inr ⟨-r, _⟩
      simp [hy]
  · rcases h with (rfl | ⟨r, rfl⟩); · simp
    by_cases hx : x = 0; · simp [hx]
    rcases lt_trichotomy r 0 with (hr | hr | hr)
    · rw [← neg_smul]
      exact Or.inr ⟨hx, smul_ne_zero hr.ne hx,
        SameRay.sameRay_pos_smul_right x (Left.neg_pos_iff.2 hr)⟩
    · simp [hr]
    · 

Proof state:
case neg.inr.inr
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x : V
r : ℝ
hx : ¬x = 0
hr : 0 < r
⊢ SameRay ℝ x (r • x) ∨ x ≠ 0 ∧ r • x ≠ 0 ∧ SameRay ℝ x (-(r • x))

Next tactic:
exact Or.inl (SameRay.sameRay_pos_smul_right x hr)
/-- The oriented angle between two vectors is not zero or `π` if and only if those two vectors
are linearly independent. -/
theorem oangle_ne_zero_and_ne_pi_iff_linearIndependent {x y : V} :
    o.oangle x y ≠ 0 ∧ o.oangle x y ≠ π ↔ LinearIndependent ℝ ![x, y] := by
  

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
⊢ oangle o x y ≠ 0 ∧ oangle o x y ≠ ↑π ↔ LinearIndependent ℝ ![x, y]

Next tactic:
rw [← not_or, ← not_iff_not, Classical.not_not,
    oangle_eq_zero_or_eq_pi_iff_not_linearIndependent]
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
⊢ x = y ↔ ‖x‖ = ‖y‖ ∧ oangle o x y = 0

Next tactic:
rw [oangle_eq_zero_iff_sameRay]
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
⊢ x = y ↔ ‖x‖ = ‖y‖ ∧ SameRay ℝ x y

Next tactic:
constructor
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · 

Proof state:
case mp
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
⊢ x = y → ‖x‖ = ‖y‖ ∧ SameRay ℝ x y

Next tactic:
rintro rfl
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · rintro rfl
    

Proof state:
case mp
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x : V
⊢ ‖x‖ = ‖x‖ ∧ SameRay ℝ x x

Next tactic:
simp
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · rintro rfl
    simp; 

Proof state:
case mp
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x : V
⊢ SameRay ℝ x x

Next tactic:
rfl
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · rintro rfl
    simp; rfl
  · 

Proof state:
case mpr
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
⊢ ‖x‖ = ‖y‖ ∧ SameRay ℝ x y → x = y

Next tactic:
rcases eq_or_ne y 0 with (rfl | hy)
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · rintro rfl
    simp; rfl
  · rcases eq_or_ne y 0 with (rfl | hy)
    · 

Proof state:
case mpr.inl
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x : V
⊢ ‖x‖ = ‖0‖ ∧ SameRay ℝ x 0 → x = 0

Next tactic:
simp
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · rintro rfl
    simp; rfl
  · rcases eq_or_ne y 0 with (rfl | hy)
    · simp
    

Proof state:
case mpr.inr
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hy : y ≠ 0
⊢ ‖x‖ = ‖y‖ ∧ SameRay ℝ x y → x = y

Next tactic:
rintro ⟨h₁, h₂⟩
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · rintro rfl
    simp; rfl
  · rcases eq_or_ne y 0 with (rfl | hy)
    · simp
    rintro ⟨h₁, h₂⟩
    

Proof state:
case mpr.inr.intro
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hy : y ≠ 0
h₁ : ‖x‖ = ‖y‖
h₂ : SameRay ℝ x y
⊢ x = y

Next tactic:
obtain ⟨r, hr, rfl⟩ := h₂.exists_nonneg_right hy
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · rintro rfl
    simp; rfl
  · rcases eq_or_ne y 0 with (rfl | hy)
    · simp
    rintro ⟨h₁, h₂⟩
    obtain ⟨r, hr, rfl⟩ := h₂.exists_nonneg_right hy
    

Proof state:
case mpr.inr.intro.intro.intro
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
y : V
hy : y ≠ 0
r : ℝ
hr : 0 ≤ r
h₁ : ‖r • y‖ = ‖y‖
h₂ : SameRay ℝ (r • y) y
⊢ r • y = y

Next tactic:
have : ‖y‖ ≠ 0 := by simpa using hy
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · rintro rfl
    simp; rfl
  · rcases eq_or_ne y 0 with (rfl | hy)
    · simp
    rintro ⟨h₁, h₂⟩
    obtain ⟨r, hr, rfl⟩ := h₂.exists_nonneg_right hy
    have : ‖y‖ ≠ 0 := by 

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
y : V
hy : y ≠ 0
r : ℝ
hr : 0 ≤ r
h₁ : ‖r • y‖ = ‖y‖
h₂ : SameRay ℝ (r • y) y
⊢ ‖y‖ ≠ 0

Next tactic:
simpa using hy
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · rintro rfl
    simp; rfl
  · rcases eq_or_ne y 0 with (rfl | hy)
    · simp
    rintro ⟨h₁, h₂⟩
    obtain ⟨r, hr, rfl⟩ := h₂.exists_nonneg_right hy
    have : ‖y‖ ≠ 0 := by simpa using hy
    

Proof state:
case mpr.inr.intro.intro.intro
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
y : V
hy : y ≠ 0
r : ℝ
hr : 0 ≤ r
h₁ : ‖r • y‖ = ‖y‖
h₂ : SameRay ℝ (r • y) y
this : ‖y‖ ≠ 0
⊢ r • y = y

Next tactic:
obtain rfl : r = 1 := by
      apply mul_right_cancel₀ this
      simpa [norm_smul, _root_.abs_of_nonneg hr] using h₁
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · rintro rfl
    simp; rfl
  · rcases eq_or_ne y 0 with (rfl | hy)
    · simp
    rintro ⟨h₁, h₂⟩
    obtain ⟨r, hr, rfl⟩ := h₂.exists_nonneg_right hy
    have : ‖y‖ ≠ 0 := by simpa using hy
    obtain rfl : r = 1 := by
      

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
y : V
hy : y ≠ 0
r : ℝ
hr : 0 ≤ r
h₁ : ‖r • y‖ = ‖y‖
h₂ : SameRay ℝ (r • y) y
this : ‖y‖ ≠ 0
⊢ r = 1

Next tactic:
apply mul_right_cancel₀ this
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · rintro rfl
    simp; rfl
  · rcases eq_or_ne y 0 with (rfl | hy)
    · simp
    rintro ⟨h₁, h₂⟩
    obtain ⟨r, hr, rfl⟩ := h₂.exists_nonneg_right hy
    have : ‖y‖ ≠ 0 := by simpa using hy
    obtain rfl : r = 1 := by
      apply mul_right_cancel₀ this
      

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
y : V
hy : y ≠ 0
r : ℝ
hr : 0 ≤ r
h₁ : ‖r • y‖ = ‖y‖
h₂ : SameRay ℝ (r • y) y
this : ‖y‖ ≠ 0
⊢ r * ‖y‖ = 1 * ‖y‖

Next tactic:
simpa [norm_smul, _root_.abs_of_nonneg hr] using h₁
/-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 := by
  rw [oangle_eq_zero_iff_sameRay]
  constructor
  · rintro rfl
    simp; rfl
  · rcases eq_or_ne y 0 with (rfl | hy)
    · simp
    rintro ⟨h₁, h₂⟩
    obtain ⟨r, hr, rfl⟩ := h₂.exists_nonneg_right hy
    have : ‖y‖ ≠ 0 := by simpa using hy
    obtain rfl : r = 1 := by
      apply mul_right_cancel₀ this
      simpa [norm_smul, _root_.abs_of_nonneg hr] using h₁
    

Proof state:
case mpr.inr.intro.intro.intro
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
y : V
hy : y ≠ 0
this : ‖y‖ ≠ 0
hr : 0 ≤ 1
h₁ : ‖1 • y‖ = ‖y‖
h₂ : SameRay ℝ (1 • y) y
⊢ 1 • y = y

Next tactic:
simp
/-- Given three nonzero vectors, the angle between the first and the second plus the angle
between the second and the third equals the angle between the first and the third. -/
@[simp]
theorem oangle_add {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x y + o.oangle y z = o.oangle x z := by
  

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ oangle o x y + oangle o y z = oangle o x z

Next tactic:
simp_rw [oangle]
/-- Given three nonzero vectors, the angle between the first and the second plus the angle
between the second and the third equals the angle between the first and the third. -/
@[simp]
theorem oangle_add {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x y + o.oangle y z = o.oangle x z := by
  simp_rw [oangle]
  

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ ↑(arg (((kahler o) x) y)) + ↑(arg (((kahler o) y) z)) = ↑(arg (((kahler o) x) z))

Next tactic:
rw [← Complex.arg_mul_coe_angle, o.kahler_mul y x z]
/-- Given three nonzero vectors, the angle between the first and the second plus the angle
between the second and the third equals the angle between the first and the third. -/
@[simp]
theorem oangle_add {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x y + o.oangle y z = o.oangle x z := by
  simp_rw [oangle]
  rw [← Complex.arg_mul_coe_angle, o.kahler_mul y x z]
  

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ ↑(arg (↑‖y‖ ^ 2 * ((kahler o) x) z)) = ↑(arg (((kahler o) x) z))
case hx
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ ((kahler o) x) y ≠ 0
case hy
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ ((kahler o) y) z ≠ 0

Next tactic:
congr 1
/-- Given three nonzero vectors, the angle between the first and the second plus the angle
between the second and the third equals the angle between the first and the third. -/
@[simp]
theorem oangle_add {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x y + o.oangle y z = o.oangle x z := by
  simp_rw [oangle]
  rw [← Complex.arg_mul_coe_angle, o.kahler_mul y x z]
  congr 1
  

Proof state:
case e_r
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ arg (↑‖y‖ ^ 2 * ((kahler o) x) z) = arg (((kahler o) x) z)
case hx
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ ((kahler o) x) y ≠ 0
case hy
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ ((kahler o) y) z ≠ 0

Next tactic:
convert Complex.arg_real_mul _ (_ : 0 < ‖y‖ ^ 2) using 2
/-- Given three nonzero vectors, the angle between the first and the second plus the angle
between the second and the third equals the angle between the first and the third. -/
@[simp]
theorem oangle_add {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x y + o.oangle y z = o.oangle x z := by
  simp_rw [oangle]
  rw [← Complex.arg_mul_coe_angle, o.kahler_mul y x z]
  congr 1
  convert Complex.arg_real_mul _ (_ : 0 < ‖y‖ ^ 2) using 2
  · 

Proof state:
case h.e'_2.h.e'_1
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ ↑‖y‖ ^ 2 * ((kahler o) x) z = ↑(‖y‖ ^ 2) * ((kahler o) x) z

Next tactic:
norm_cast
/-- Given three nonzero vectors, the angle between the first and the second plus the angle
between the second and the third equals the angle between the first and the third. -/
@[simp]
theorem oangle_add {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x y + o.oangle y z = o.oangle x z := by
  simp_rw [oangle]
  rw [← Complex.arg_mul_coe_angle, o.kahler_mul y x z]
  congr 1
  convert Complex.arg_real_mul _ (_ : 0 < ‖y‖ ^ 2) using 2
  · norm_cast
  · 

Proof state:
case e_r.convert_2
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ 0 < ‖y‖ ^ 2

Next tactic:
have : 0 < ‖y‖ := by simpa using hy
/-- Given three nonzero vectors, the angle between the first and the second plus the angle
between the second and the third equals the angle between the first and the third. -/
@[simp]
theorem oangle_add {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x y + o.oangle y z = o.oangle x z := by
  simp_rw [oangle]
  rw [← Complex.arg_mul_coe_angle, o.kahler_mul y x z]
  congr 1
  convert Complex.arg_real_mul _ (_ : 0 < ‖y‖ ^ 2) using 2
  · norm_cast
  · have : 0 < ‖y‖ := by 

Proof state:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ 0 < ‖y‖

Next tactic:
simpa using hy