ofAlgEquiv_algEquiv — Mathlib

English

Compatibility of ofAlgEquiv with algEquiv: for h'' and e, the two formulations of the induced isomorphisms agree up to equality.

Русский

Совместимость ofAlgEquiv с algEquiv: два формулирования эквивалентны.

LaTeX

$$IsAdjoinRoot.ofAlgEquiv_algEquiv$$

Lean4

@[simp]
theorem ofAlgEquiv_algEquiv {U : Type*} [Ring U] [Algebra R U] (h'' : IsAdjoinRoot U f) (e : S ≃ₐ[R] T) :
    (h.ofAlgEquiv e).algEquiv h'' = e.symm.trans (h.algEquiv h'') :=
  by
  ext a
  simp_rw [algEquiv_def, AlgEquiv.trans_apply, EmbeddingLike.apply_eq_iff_eq, AlgEquiv.symm_apply_eq,
    adjoinRootAlgEquiv_apply_eq_map, ofAlgEquiv_map_apply, ← adjoinRootAlgEquiv_apply_eq_map, AlgEquiv.apply_symm_apply]

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