restrictRestrictAlgEquivMapHom_injective

English

If K and L generate E (K ⊔ L = ⊤), then restriction from Aut_L(E) to Aut_K(E) is injective.

Русский

Если K и L порождают E (K ⊔ L = ⊤), то ограничение от Aut_L(E) к Aut_K(E) эез injective.

LaTeX

$$$K\sqcup L = \top \Rightarrow \operatorname{RestrictRestrictAlgEquivMapHom}_{F}(K,L,E) \text{ is injective}$$$

Lean4

theorem restrictRestrictAlgEquivMapHom_injective (h : K ⊔ L = ⊤) :
    Function.Injective (restrictRestrictAlgEquivMapHom F K L E) :=
  by
  refine (injective_iff_map_eq_one _).mpr fun φ hφ ↦ ?_
  suffices h : MulSemiringAction.toAlgAut (E ≃ₐ[↥L] E) F E φ = 1 by rwa [AlgEquiv.ext_iff] at h ⊢
  rw [← Subgroup.mem_bot, ← fixingSubgroup_top, ← h, fixingSubgroup_sup]
  exact ⟨fun x ↦ (hφ ▸ restrictRestrictAlgEquivMapHom_apply K L φ x).symm, φ.commutes⟩

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