/-- A family is affinely independent if and only if any affine
combinations (with sum of weights 1) that evaluate to the same point
have equal `Set.indicator`. -/
theorem affineIndependent_iff_indicator_eq_of_affineCombination_eq (p : ι → P) :
    AffineIndependent k p ↔
      ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),
        ∑ i in s1, w1 i = 1 →
          ∑ i in s2, w2 i = 1 →
            s1.affineCombination k p w1 = s2.affineCombination k p w2 →
              Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2 := by
  classical
    constructor
    · intro ha s1 s2 w1 w2 hw1 hw2 heq
      ext i
      by_cases hi : i ∈ s1 ∪ s2
      · rw [← sub_eq_zero]
        rw [← Finset.sum_indicator_subset _ (Finset.subset_union_left s1 s2)] at hw1
        rw [← Finset.sum_indicator_subset _ (Finset.subset_union_right s1 s2)] at hw2
        have hws : (∑ i in s1 ∪ s2, (Set.indicator (↑s1) w1 - Set.indicator (↑s2) w2) i) = 0 := by
          simp [hw1, hw2]
        rw [Finset.affineCombination_indicator_subset _ _ (Finset.subset_union_left s1 s2),
          Finset.affineCombination_indicator_subset _ _ (Finset.subset_union_right s1 s2),
          ← @vsub_eq_zero_iff_eq V, Finset.affineCombination_vsub] at heq
        exact ha (s1 ∪ s2) (Set.indicator (↑s1) w1 - Set.indicator (↑s2) w2) hws heq i hi
      · rw [← Finset.mem_coe, Finset.coe_union] at hi
        have h₁ : Set.indicator (↑s1) w1 i = 0 := by
          simp only [Set.indicator, Finset.mem_coe, ite_eq_right_iff]
          intro h
          by_contra
          exact (mt (@Set.mem_union_left _ i ↑s1 ↑s2) hi) h
        have h₂ : Set.indicator (↑s2) w2 i = 0 := by
          simp only [Set.indicator, Finset.mem_coe, ite_eq_right_iff]
          intro h
          by_contra
          exact (mt (@Set.mem_union_right _ i ↑s2 ↑s1) hi) h
        simp [h₁, h₂]
    · intro ha s w hw hs i0 hi0
      let w1 : ι → k := Function.update (Function.const ι 0) i0 1
      have hw1 : ∑ i in s, w1 i = 1 := by
        rw [Finset.sum_update_of_mem hi0]
        simp only [Finset.sum_const_zero, add_zero, const_apply]
      have hw1s : s.affineCombination k p w1 = p i0 :=
        s.affineCombination_of_eq_one_of_eq_zero w1 p hi0 (Function.update_same _ _ _)
          fun _ _ hne => Function.update_noteq hne _ _
      let w2 := w + w1
      have hw2 : ∑ i in s, w2 i = 1 := by
        simp_all only [Pi.add_apply, Finset.sum_add_distrib, zero_add]
      have hw2s : s.affineCombination k p w2 = p i0 := by
        simp_all only [← Finset.weightedVSub_vadd_affineCombination, zero_vadd]
      replace ha := ha s s w2 w1 hw2 hw1 (hw1s.symm ▸ hw2s)
      have hws : w2 i0 - w1 i0 = 0 := by
        rw [← Finset.mem_coe] at hi0
        rw [← Set.indicator_of_mem hi0 w2, ← Set.indicator_of_mem hi0 w1, ha, sub_self]
      

Proof state:
case mpr
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
s : Finset ι
w : ι → k
hw : ∑ i in s, w i = 0
hs : (weightedVSub s p) w = 0
i0 : ι
hi0 : i0 ∈ s
w1 : ι → k := update (const ι 0) i0 1
hw1 : ∑ i in s, w1 i = 1
hw1s : (affineCombination k s p) w1 = p i0
w2 : ι → k := w + w1
hw2 : ∑ i in s, w2 i = 1
hw2s : (affineCombination k s p) w2 = p i0
ha : Set.indicator (↑s) w2 = Set.indicator (↑s) w1
hws : w2 i0 - w1 i0 = 0
⊢ w i0 = 0

Next tactic:
simpa using hws