eval₂_pUnitAlgEquiv_symm — Mathlib

English

For any algebra equivalence e, the inverse of mapAlgEquiv σ e equals mapAlgEquiv σ e.symm.

Русский

Для любой эквивалентности e, обратное отображение mapAlgEquiv σ e равно mapAlgEquiv σ e.symm.

LaTeX

$$$$ (mapAlgEquiv \\; σ\\; e)^{-1} = mapAlgEquiv \\; σ\\; e^{-1}. $$$$

Lean4

theorem eval₂_pUnitAlgEquiv_symm {f : Polynomial R} {φ : R →+* S} {a : Unit → S} :
    ((MvPolynomial.pUnitAlgEquiv R).symm f : MvPolynomial Unit R).eval₂ φ a = f.eval₂ φ (a ()) :=
  by
  simp only [MvPolynomial.pUnitAlgEquiv_symm_apply]
  induction f using Polynomial.induction_on' with
  | add f g hf hg => simp [hf, hg]
  | monomial n r => simp

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