eqId_iff_eq — Mathlib

English

Equality-equivalence δ: A.EqId ⇔ (Opposite.unop A.fst).len = (Opposite.unop Δ).len.

Русский

Эквивалентность EqId по длине: A.EqId ⇔ (Opposite.unop A.fst).len = (Opposite.unop Δ).len.

LaTeX

$$$A\\mathrm{EqId} \\Leftrightarrow (\\mathrm{Opposite.unop}\\,A.fst).\\mathrm{len} = (\\mathrm{Opposite.unop}\\,Δ).\\mathrm{len}$$$

Lean4

theorem eqId_iff_eq : A.EqId ↔ A.1 = Δ := by
  constructor
  · intro h
    dsimp at h
    rw [h]
    rfl
  · intro h
    rcases A with ⟨_, ⟨f, hf⟩⟩
    simp only at h
    subst h
    refine ext _ _ rfl ?_
    simp only [eqToHom_refl, comp_id]
    exact eq_id_of_epi f

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