d_comp_d — Mathlib

English

The differentials compose to zero: d_n ∘ d_{n+1} = 0 for all n.

Русский

Союз дифференциалов даёт ноль: d_n ∘ d_{n+1} = 0 для всех n.

LaTeX

$$$d_n \\circ d_{n+1} = 0$ for all n$$

Lean4

@[reassoc (attr := simp)]
theorem d_comp_d (n : ℕ) : d R G n ≫ d R G (n + 1) = 0 := by
  induction n with
  | zero =>
    rw [d_succ, Preadditive.comp_sub, sub_eq_zero]
    rfl
  | succ n ih =>
    rw [d_succ R G (n + 1), Preadditive.comp_sub]
    nth_rw 2 [d_succ]
    rw [Preadditive.sub_comp, ← whiskerRight_comp, ih, Functor.whiskerRight_zero, sub_zero, sub_eq_zero]
    rfl

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