touchpointWeights_eq_zero — Mathlib

English

Weights for the empty signs configuration coincide with themselves and ensure consistency.

Русский

Вес касательных точек при пустом наборе знаков согласуется с весами касательных точек при пустой конфигурации знаков.

LaTeX

$$$ s.touchpointWeights(\emptyset, i, j) = s.touchpointWeights(\emptyset, i, j) $$$

Lean4

@[simp]
theorem touchpointWeights_eq_zero {signs : Finset (Fin (n + 1))} (i : Fin (n + 1)) :
    s.touchpointWeights signs i i = 0 :=
  by
  refine
    s.independent.eq_zero_of_affineCombination_mem_affineSpan (s.sum_touchpointWeights signs i) ?_ (Finset.mem_univ _)
      (Set.notMem_compl_iff.2 (Set.mem_singleton _))
  rw [s.affineCombination_touchpointWeights]
  convert s.touchpoint_mem_affineSpan _ _
  simp

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