English
If hf is a left inverse of algebraMap R S and quasispectrum S a ⊆ range (algebraMap R S), then a QuasispectrumRestricts a f holds.
Русский
Если hf — левый обратный к algebraMap R S и quasispectrum S a ⊆ range (algebraMap R S), то выполняется QuasispectrumRestricts a f.
LaTeX
$$QuasispectrumRestricts a f$$
Lean4
theorem quasispectrumRestricts_iff {R S A : Type*} [CommSemiring R] [CommSemiring S] [NonUnitalRing A] [Module R A]
[Module S A] [Algebra R S] (a : A) (f : S → R) :
QuasispectrumRestricts a f ↔
(quasispectrum S a).RightInvOn f (algebraMap R S) ∧ Function.LeftInverse f (algebraMap R S) :=
⟨fun ⟨h₁, h₂⟩ ↦ ⟨h₁, h₂⟩, fun ⟨h₁, h₂⟩ ↦ ⟨h₁, h₂⟩⟩