ofMkLEMk_comp_ofMkLEMk — Mathlib

English

Composition of two MkLE maps followed by another yields MkLE for the transitive inclusion.

Русский

Сложение MkLE–карт приводит к MkLE для переходящего включения.

LaTeX

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

Lean4

@[reassoc (attr := simp)]
theorem ofMkLEMk_comp_ofMkLEMk {B A₁ A₂ A₃ : C} (f : A₁ ⟶ B) [Mono f] (g : A₂ ⟶ B) [Mono g] (h : A₃ ⟶ B) [Mono h]
    (h₁ : mk f ≤ mk g) (h₂ : mk g ≤ mk h) : ofMkLEMk f g h₁ ≫ ofMkLEMk g h h₂ = ofMkLEMk f h (h₁.trans h₂) :=
  by
  simp only [ofLE, ofMkLEMk, ← Functor.map_comp_assoc underlying, assoc, Iso.hom_inv_id_assoc]
  congr 1

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