comp — Mathlib

English

Simp lemma for the Jacobian in the bijective algebra map case.

Русский

Лемма-упрощение для Якобиана в случае биективной алгебраической карты.

LaTeX

$$simp [jacobian_eq_jacobiMatrix_det, RingHom.map_det, Matrix.det_one]$$

Lean4

/-- Given an `R`-algebra `S` and an `S`-algebra `T` with pre-submersive presentations,
this is the canonical pre-submersive presentation of `T` as an `R`-algebra. -/
@[simps map]
noncomputable def comp : PreSubmersivePresentation R T (ι' ⊕ ι) (σ' ⊕ σ)
    where
  __ := Q.toPresentation.comp P.toPresentation
  map := Sum.elim (fun rq ↦ Sum.inl <| Q.map rq) (fun rp ↦ Sum.inr <| P.map rp)
  map_inj :=
    Function.Injective.sumElim ((Sum.inl_injective).comp (Q.map_inj)) ((Sum.inr_injective).comp (P.map_inj)) <| by simp

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