limitsOfShape_isoClosure — Mathlib

English

Limits commute with iso-closure: P.isoClosure.limitsOfShape J = P.limitsOfShape J.

Русский

Пределы коммутируют с iso-closure: P.isoClosure.limitsOfShape J = P.limitsOfShape J.

LaTeX

$$Similarly to 78064, but stated as a theorem about limits under iso-closure.$$

Lean4

@[simp]
theorem limitsOfShape_isoClosure : P.isoClosure.limitsOfShape J = P.limitsOfShape J :=
  by
  refine le_antisymm ?_ (limitsOfShape_monotone _ (P.le_isoClosure))
  intro X ⟨h⟩
  choose obj h₁ h₂ using h.prop_diag_obj
  exact
    ⟨{  toLimitPresentation := h.changeDiag (h.diag.isoCopyObj obj (fun j ↦ (h₂ j).some)).symm
        prop_diag_obj := h₁ }⟩

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