apply_linearCombination' — Mathlib

English

A bilinear form is separating left if its associated quadratic form is anisotropic.

Русский

Билинейная форма, которая отделяет слева, если ассоциированная квадратичная форма анизотропна.

LaTeX

$$separatingLeft_of_anisotropic {B} (hB : B.toQuadraticMap.Anisotropic) : B.SeparatingLeft$$

Lean4

theorem apply_linearCombination' (Q : QuadraticMap R M N) {g : ι → M} (l : ι →₀ R) :
    Q (linearCombination R g l) =
      linearCombination R (polarSym2 Q ∘ Sym2.map g) l.sym2Mul - linearCombination R (Q ∘ g) (l * l) :=
  by
  simp_rw [linearCombination_apply, map_finsuppSum', Q.map_smul, mul_smul]
  rw [(l * l).sum_of_support_subset support_mul_subset_left _ <| by simp,
    l.sym2Mul.sum_of_support_subset support_sym2Mul_subset _ <| by simp]
  simp [Finsupp.sum, ← polarSym2_map_smul, mul_smul]

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