of_equiv — Mathlib

English

The pushout isomorphism respects the left and right algebra maps through tmul and algebra maps.

Русский

Изоморфизм пушаута уважает левые и правые отображения через tmul и алгебра map.

LaTeX

$$Algebra.IsPushout.equiv, Algebra.IsPushout.equiv_tmul$$

Lean4

theorem of_equiv [h : IsPushout R R' S S'] {T : Type*} [CommSemiring T] [Algebra R' T] [Algebra S T] [Algebra R T]
    [IsScalarTower R S T] [IsScalarTower R R' T] (e : S' ≃ₐ[R'] T)
    (he : e.toRingHom.comp (algebraMap S S') = algebraMap S T) : IsPushout R R' S T :=
  by
  rw [isPushout_iff] at h ⊢
  refine IsBaseChange.of_equiv (h.equiv ≪≫ₗ e.toLinearEquiv) fun x ↦ ?_
  simpa [h.equiv_tmul] using DFunLike.congr_fun he x

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