inner_mul_symm_re_eq_norm — Mathlib

English

⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫.

Русский

⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫.

LaTeX

$$$\\\\langle x - y, x - y \\\\rangle = \\\\langle x, x \\\\rangle - \\\\langle x, y \\\\rangle - \\\\langle y, x \\\\rangle + \\\\langle y, y \\\\rangle$$$

Lean4

theorem inner_mul_symm_re_eq_norm (x y : F) : re (⟪x, y⟫ * ⟪y, x⟫) = ‖⟪x, y⟫ * ⟪y, x⟫‖ :=
  by
  rw [← inner_conj_symm, mul_comm]
  exact re_eq_norm_of_mul_conj ⟪y, x⟫

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