inner_vsub_right_eq_zero_symm — Mathlib

English

For a unitary operator u, inner products are preserved: ⟪u x, u y⟫ = ⟪x, y⟫ for all x,y.

Русский

Унитарный оператор сохраняет скалярное произведение: ⟪u x, u y⟫ = ⟪x, y⟫ для любых x,y.

LaTeX

$$$\langle u x, u y\rangle = \langle x, y\rangle$ for all x,y$$

Lean4

theorem inner_vsub_right_eq_zero_symm {v : V} {a b : P} : ⟪v, a -ᵥ b⟫_𝕜 = 0 ↔ ⟪v, b -ᵥ a⟫_𝕜 = 0 := by
  rw [← neg_vsub_eq_vsub_rev, inner_neg_right, neg_eq_zero]

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