ofMkLE_comp_ofLEMk — Mathlib

English

The composition of ofMkLE and ofLE equals the ofMkLE for the transitive inclusion mk f ≤ X ≤ Y.

Русский

Сложение ofMkLE и ofLE даёт ofMkLE для переходящего включения mk f ≤ X ≤ Y.

LaTeX

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

Lean4

@[reassoc (attr := simp)]
theorem ofMkLE_comp_ofLEMk {B A₁ A₂ : C} (f : A₁ ⟶ B) [Mono f] (X : Subobject B) (g : A₂ ⟶ B) [Mono g] (h₁ : mk f ≤ X)
    (h₂ : X ≤ mk g) : ofMkLE f X h₁ ≫ ofLEMk X g h₂ = ofMkLEMk f g (h₁.trans h₂) :=
  by
  simp only [ofMkLE, ofLEMk, ofLE, ofMkLEMk, ← Functor.map_comp_assoc underlying, assoc]
  congr 1

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