orthonormal_of_forall_eq_or_eq_neg

English

Repeat: the zero inner product holds under the support exclusion condition for l.

Русский

Повторение: нулевое скалярное произведение сохраняется при условии исключения индекса из опоры.

LaTeX

$$$i \notin s \Rightarrow \langle Finsupp.linearCombination 𝕜 v l, v i \rangle = 0$$$

Lean4

/-- Given an orthonormal family, a second family of vectors is orthonormal if every vector equals
the corresponding vector in the original family or its negation. -/
theorem orthonormal_of_forall_eq_or_eq_neg {v w : ι → E} (hv : Orthonormal 𝕜 v) (hw : ∀ i, w i = v i ∨ w i = -v i) :
    Orthonormal 𝕜 w := by
  classical
  rw [orthonormal_iff_ite] at *
  intro i j
  rcases hw i with hi | hi <;> rcases hw j with hj | hj <;> replace hv := hv i j <;> split_ifs at hv ⊢ with h <;>
    simpa only [hi, hj, h, inner_neg_right, inner_neg_left, neg_neg, eq_self_iff_true, neg_eq_zero] using hv

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