toSubmonoid_closure — Mathlib

English

Let S : ι → Submonoid M^{op}. Then (iInf S).unop = ⨅ i, (S i).unop.

Русский

Пусть S : ι → подмономоды M^{op}. Тогда (iInf S).unop = ⨅ i, (S i).unop.

LaTeX

$$$ (\iInf S).unop = \big\wedge_{i} (S(i)).unop $$$

Lean4

theorem toSubmonoid_closure (S : Set A) :
    (AddSubmonoid.toSubmonoid) (AddSubmonoid.closure S) = Submonoid.closure (Multiplicative.toAdd ⁻¹' S) :=
  le_antisymm
    (AddSubmonoid.toSubmonoid.to_galoisConnection.l_le <|
      AddSubmonoid.closure_le.2 <| Submonoid.subset_closure (M := Multiplicative A))
    (Submonoid.closure_le.2 <| AddSubmonoid.subset_closure (M := A))

Похожие утверждения