liftFromProjective_comp — Mathlib

English

If hS is exact and P is projective, then hS.liftFromProjective f hf ≫ f = f when f ≫ S.g = 0.

Русский

Если hS точен и P проектный, то hS.liftFromProjective f hf затем f равно f при условии f ≫ S.g = 0.

LaTeX

$$$hS.liftFromProjective f hf \;\; \Rightarrow \, hS.liftFromProjective f hf \;\; f = f$$$

Lean4

@[reassoc (attr := simp, nolint unusedHavesSuffices)]
theorem liftFromProjective_comp (hS : S.Exact) {P : C} (f : P ⟶ S.X₂) [Projective P] (hf : f ≫ S.g = 0) :
    hS.liftFromProjective f hf ≫ S.f = f := by
  dsimp [liftFromProjective]
  rw [← toCycles_i, Projective.factorThru_comp_assoc, liftCycles_i]

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