-- Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
theorem sin_angle_add_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖

Proof state:
case neg.h
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ≠ 0

Tactic:
refine (ne_of_lt ?_).symm

Result:
case neg.h
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ 0 < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

Proof state:
case neg.h
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ ‖x + y‖ ≠ 0

Tactic:
rw [← mul_self_ne_zero, norm_add_sq_eq_norm_sq_add_norm_sq_real h]

Result:
case neg.h
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ≠ 0

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬(x = 0 ∧ y = 0)
⊢ Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖

Tactic:
rw [not_and_or] at h0

Result:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖

Proof state:
case neg.h.inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬x = 0
⊢ 0 < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

Tactic:
exact Left.add_pos_of_pos_of_nonneg (mul_self_pos.2 (norm_ne_zero_iff.2 h0)) (mul_self_nonneg _)

Result:
no goals

Proof state:
case pos
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : x = 0 ∧ y = 0
⊢ Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖

Tactic:
simp [h0]

Result:
no goals

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖

Tactic:
rw [sin_angle_add_of_inner_eq_zero h h0, div_mul_cancel₀]

Result:
case neg.h
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ ‖x + y‖ ≠ 0

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
⊢ Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖

Tactic:
by_cases h0 : x = 0 ∧ y = 0

Result:
case pos
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : x = 0 ∧ y = 0
⊢ Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖

case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬(x = 0 ∧ y = 0)
⊢ Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖

Proof state:
case neg.h.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬y = 0
⊢ 0 < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

Tactic:
exact Left.add_pos_of_nonneg_of_pos (mul_self_nonneg _) (mul_self_pos.2 (norm_ne_zero_iff.2 h0))

Result:
no goals

Proof state:
case neg.h
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬x = 0 ∨ ¬y = 0
⊢ 0 < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

Tactic:
rcases h0 with (h0 | h0)

Result:
case neg.h.inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬x = 0
⊢ 0 < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

case neg.h.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : ¬y = 0
⊢ 0 < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖