liftK_π_eq_zero_of_boundary — Mathlib

English

For every A, morphisms k : A → X2 and x : A → X1 with k = x ∘ f, the lifted morphism satisfies h.liftK k hx ≫ h.π = 0.

Русский

Для всякого A, морфизмов k : A → X2 и x : A → X1 с условием k = x ∘ f, возведённый морфизм удовлетворяет h.liftK k hx ≫ h.π = 0.

LaTeX

$$$ \\forall A,\\ k : A \\to S.X_2,\\ x : A \\to S.X_1,\\ hx : k = x \\circ S.f,\\ h.liftK k (by hx) \\;\\gg\\; h.π = 0 $$$

Lean4

@[reassoc]
theorem liftK_π_eq_zero_of_boundary (k : A ⟶ S.X₂) (x : A ⟶ S.X₁) (hx : k = x ≫ S.f) :
    h.liftK k (by rw [hx, assoc, S.zero, comp_zero]) ≫ h.π = 0 :=
  by
  rw [show 0 = (x ≫ h.f') ≫ h.π by simp]
  congr 1
  simp only [← cancel_mono h.i, hx, liftK_i, assoc, f'_i]

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