English
The basicOpen construction is functorial with respect to restriction of opens.
Русский
Конструкция basicOpen относимa к ограничениям открытых множеств по отношению к отображениям.
LaTeX
$$$\text{basicOpen}_{X}(f)|_{i} = \text{basicOpen}_{X}(f|_{i})$$$
Lean4
@[simp]
theorem basicOpen_res {U V : (Opens X)ᵒᵖ} (i : U ⟶ V) (f : X.presheaf.obj U) :
@basicOpen X (unop V) (X.presheaf.map i f) = unop V ⊓ @basicOpen X (unop U) f :=
by
ext x; constructor
· rintro ⟨hxV, hx⟩
rw [germ_res_apply' X.presheaf] at hx
exact ⟨hxV, i.unop.le hxV, hx⟩
· rintro ⟨hxV, _, hx⟩
refine ⟨hxV, ?_⟩
rw [germ_res_apply' X.presheaf]
exact hx