Theorem: MvPolynomial.quotientEquivQuotientMvPolynomial_leftInverse
File: Mathlib/RingTheory/Polynomial/Quotient.lean

Proof steps:
state_before: R : Type u_1
σ : Type u_2
inst✝ : CommRing R
r : R
I : Ideal R
⊢ Function.LeftInverse
    (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯) fun i =>
      (Ideal.Quotient.mk (Ideal.map C I)) (X i))
    ⇑(Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)
state_after: R : Type u_1
σ : Type u_2
inst✝ : CommRing R
r : R
I : Ideal R
f : MvPolynomial σ R ⧸ Ideal.map C I
⊢ eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)
      (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i))
      ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯) f) =
    f
tactic: intro f
state_before: R : Type u_1
σ : Type u_2
inst✝ : CommRing R
r : R
I : Ideal R
f : MvPolynomial σ R ⧸ Ideal.map C I
⊢ eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)
      (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i))
      ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯) f) =
    f
state_after: case intro
R : Type u_1
σ : Type u_2
inst✝ : CommRing R
r : R
I : Ideal R
f : MvPolynomial σ R
⊢ eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)
      (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i))
      ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)
        ((Ideal.Quotient.mk (Ideal.map C I)) f)) =
    (Ideal.Quotient.mk (Ideal.map C I)) f
tactic: obtain ⟨f, rfl⟩ := Ideal.Quotient.mk_surjective f
state_before: case intro
R : Type u_1
σ : Type u_2
inst✝ : CommRing R
r : R
I : Ideal R
f : MvPolynomial σ R
⊢ eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)
      (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i))
      ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)
        ((Ideal.Quotient.mk (Ideal.map C I)) f)) =
    (Ideal.Quotient.mk (Ideal.map C I)) f
state_after: case intro.h_C
R : Type u_1
σ : Type u_2
inst✝ : CommRing R
r : R
I : Ideal R
f : MvPolynomial σ R
⊢ ∀ (a : R),
    eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)
        (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i))
        ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)
          ((Ideal.Quotient.mk (Ideal.map C I)) (C a))) =
      (Ideal.Quotient.mk (Ideal.map C I)) (C a)

case intro.h_add
R : Type u_1
σ : Type u_2
inst✝ : CommRing R
r : R
I : Ideal R
f : MvPolynomial σ R
⊢ ∀ (p q : MvPolynomial σ R),
    eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)
          (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i))
          ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)
            ((Ideal.Quotient.mk (Ideal.map C I)) p)) =
        (Ideal.Quotient.mk (Ideal.map C I)) p →
      eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)
            (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i))
            ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)
              ((Ideal.Quotient.mk (Ideal.map C I)) q)) =
          (Ideal.Quotient.mk (Ideal.map C I)) q →
        eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)
            (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i))
            ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)
              ((Ideal.Quotient.mk (Ideal.map C I)) (p + q))) =
          (Ideal.Quotient.mk (Ideal.map C I)) (p + q)

case intro.h_X
R : Type u_1
σ : Type u_2
inst✝ : CommRing R
r : R
I : Ideal R
f : MvPolynomial σ R
⊢ ∀ (p : MvPolynomial σ R) (n : σ),
    eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)
          (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i))
          ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)
            ((Ideal.Quotient.mk (Ideal.map C I)) p)) =
        (Ideal.Quotient.mk (Ideal.map C I)) p →
      eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)
          (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i))
          ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)
            ((Ideal.Quotient.mk (Ideal.map C I)) (p * X n))) =
        (Ideal.Quotient.mk (Ideal.map C I)) (p * X n)
tactic: apply induction_on f