quotIdealMapEquivTensorQuot — Mathlib

English

The snd component of prodRight applied to a ⊗ x is a ⊗ x.2.

Русский

Для snd-компоненты prodRight применённой к a ⊗ x равен a ⊗ x.2.

LaTeX

$$prodRight_tmul_snd$$

Lean4

/-- `B ⊗[A] (A ⧸ I)` is isomorphic as an `A`-algebra to `B ⧸ I B`. -/
noncomputable def quotIdealMapEquivTensorQuot : (B ⧸ (I.map <| algebraMap A B)) ≃ₐ[B] B ⊗[A] (A ⧸ I) :=
  AlgEquiv.ofLinearEquiv (quotIdealMapEquivTensorQuotAux B I) rfl
    (fun x y ↦ by
      obtain ⟨u, rfl⟩ := Ideal.Quotient.mk_surjective x
      obtain ⟨v, rfl⟩ := Ideal.Quotient.mk_surjective y
      simp_rw [← map_mul, quotIdealMapEquivTensorQuotAux_mk]
      simp)

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