English
The star operation commutes with the algebraic adjoin: star(Algebra.adjoin(R,s)) = Algebra.adjoin(R, star s).
Русский
Звездовая операция commute c алгебраическим дополнением: star(Algebra.adjoin(R,s)) = Algebra.adjoin(R, star s).
LaTeX
$$$\\star(\\mathrm{Algebra.adjoin}\\ (R\\, s)) = \\mathrm{Algebra.adjoin}\\ (R\\, (\\mathrm{star}\,s))$$$
Lean4
/-- The star operation on `Subalgebra` commutes with `Algebra.adjoin`. -/
theorem star_adjoin_comm (s : Set A) : star (Algebra.adjoin R s) = Algebra.adjoin R (star s) :=
have this : ∀ t : Set A, Algebra.adjoin R (star t) ≤ star (Algebra.adjoin R t) := fun _ =>
Algebra.adjoin_le fun _ hx => Algebra.subset_adjoin hx
le_antisymm (by simpa only [star_star] using Subalgebra.star_mono (this (star s))) (this s)
