ofLE_comp_ofLEMk — Mathlib

English

The composition of an ofLE map with an ofLEMk map equals the ofLEMk map for the transitive composition of subobjects.

Русский

Сложение ofLE и ofLEMk равно to mk движению по трём подобъектам: h1.trans h2.

LaTeX

$$$\operatorname{ofLE} X Y h_1 \;\circ\; \operatorname{ofLEMk} Y f h_2 = \operatorname{ofLEMk} X f (h_1 \trans h_2)$$$

Lean4

@[reassoc (attr := simp)]
theorem ofLE_comp_ofLEMk {B A : C} (X Y : Subobject B) (f : A ⟶ B) [Mono f] (h₁ : X ≤ Y) (h₂ : Y ≤ mk f) :
    ofLE X Y h₁ ≫ ofLEMk Y f h₂ = ofLEMk X f (h₁.trans h₂) :=
  by
  simp only [ofLEMk, ofLE, ← Functor.map_comp_assoc underlying]
  congr 1

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