openOfElementHom — Mathlib

English

There is a canonical frame homomorphism from L to the lattice of sets of points of L given by u ↦ {x | x u}, preserving inf, top, and sup.

Русский

Существует канонический гомоморфизм рам from L в множество подмножеств PT L, заданный u ↦ {x | x u}, сохраняющий inf, top и sup.

LaTeX

$$$\mathrm{openOfElementHom} : \mathrm{FrameHom}\,L\; (\mathcal{P}(\mathrm{PT}L))$ with toFun(u)=\{x\nmid x(u)\}$ and preserves inf, top, and sSup$$

Lean4

/-- The frame homomorphism from a complete lattice `L` to the complete lattice of sets of
points of `L`. -/
@[simps]
def openOfElementHom : FrameHom L (Set (PT L))
    where
  toFun u := {x | x u}
  map_inf' a b := by simp [Set.setOf_and]
  map_top' := by simp
  map_sSup' S := by ext; simp [«Prop».exists_iff]

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