compClosure_iff_self — Mathlib

English

Assuming r is a Congruence, CompClosure r is equal to r; i.e., two morphisms are equivalent under CompClosure if and only if they are r-related.

Русский

Пусть r — конгруэнция; CompClosure r равен r; две гомоморфизмы эквивалентны по CompClosure тогда и только тогда, когда они связаны через r.

LaTeX

$$$\mathrm{CompClosure}(r) = r$ under the assumption that r is a Congruence; i.e., \forall f,g:\, X\to Y, \mathrm{CompClosure}(r,f,g) \iff r\ f\ g.$$$

Lean4

theorem compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) : CompClosure r f g ↔ r f g :=
  by
  constructor
  · rintro ⟨hfg⟩
    exact Congruence.compLeft _ (Congruence.compRight _ (by assumption))
  · exact CompClosure.of _ _ _

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