English
For a subcomplex A of X and a fixed n, the degenerate n-simplices of A are top if and only if the intersection of X's degenerate n-simplices with A.obj n equals A.obj n.
Русский
Для подслоя A подмножество X и фиксированного n degenerate n-сгенератов A является верхним тогда и только тогда пересечение degenerate n из X и A.obj n равно A.obj n.
LaTeX
$$$$\degenerate A n = \top \iff (X.degenerate n \cap A.obj _) = A.obj _,$$$$
Lean4
theorem degenerate_eq_top_iff (n : ℕ) : degenerate A n = ⊤ ↔ (X.degenerate n ⊓ A.obj _) = A.obj _ :=
by
constructor
· intro h
ext x
simp only [Set.inf_eq_inter, Set.mem_inter_iff, and_iff_right_iff_imp]
intro hx
simp [← A.mem_degenerate_iff ⟨x, hx⟩, h, Set.top_eq_univ, Set.mem_univ]
· intro h
simp only [Set.inf_eq_inter, Set.inter_eq_right] at h
ext x
simpa [A.mem_degenerate_iff] using h x.prop
