English
If L is algebraic over K and R is a DivisionRing with compatible K-structures and a tower K ⊆ L ⊆ R, then any K-algebra hom from L to R is bijective.
Русский
Если L алгебраичен над K и существует совместимость структур и тетрадаповая башня K ⊆ L ⊆ R, то любой K-алгебраический однород из L в R биективен.
LaTeX
$$$[\mathrm{NoZeroSMulDivisors\ K\ L}] [\mathrm{Algebra.IsAlgebraic\ K\ L}] [\mathrm{DivisionRing\ R}] [\mathrm{Algebra\ K\ R}] [\mathrm{Algebra\ L\ R}] [\mathrm{IsScalarTower\ K\ L\ R}] (f : L \to_{K} R) : \mathrm{Function.Bijective}\ f$$$
Lean4
theorem algHom_bijective₂ [NoZeroSMulDivisors K L] [DivisionRing R] [Algebra K R] [Algebra.IsAlgebraic K L]
(f : L →ₐ[K] R) (g : R →ₐ[K] L) : Function.Bijective f ∧ Function.Bijective g :=
(g.injective.bijective₂_of_surjective f.injective (algHom_bijective <| g.comp f).2).symm
