ofLEMk_comp_ofMkLE — Mathlib

English

The composition of ofLEMk and ofMkLE along mk f and Y yields the LE of X to Y: ofLEMk X f h1 ≫ ofMkLE f Y h2 = ofLE X Y (h1.trans h2).

Русский

Сложение ofLEMk и ofMkLE даёт ofLE между X и Y через h1.trans h2.

LaTeX

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

Lean4

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

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