Theorem: Ideal.eq_zero_of_polynomial_mem_map_range
File: Mathlib/RingTheory/Polynomial/Quotient.lean

Proof steps:
state_before: R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
x : ↥((Quotient.mk I).comp C).range
hx : C x ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
⊢ x = 0
state_after: R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
x : ↥((Quotient.mk I).comp C).range
hx : C x ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
⊢ x = 0
tactic: let i := ((Quotient.mk I).comp C).rangeRestrict
state_before: R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
x : ↥((Quotient.mk I).comp C).range
hx : C x ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
⊢ x = 0
state_after: R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
x : ↥((Quotient.mk I).comp C).range
hx : C x ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
⊢ x = 0
tactic: have hi' : RingHom.ker (Polynomial.mapRingHom i) ≤ I := by
    refine fun f hf => polynomial_mem_ideal_of_coeff_mem_ideal I f fun n => ?_
    rw [mem_comap, ← Quotient.eq_zero_iff_mem, ← RingHom.comp_apply]
    rw [RingHom.mem_ker, coe_mapRingHom] at hf
    replace hf := congr_arg (fun f : Polynomial _ => f.coeff n) hf
    simp only [coeff_map, coeff_zero] at hf
    rwa [Subtype.ext_iff, RingHom.coe_rangeRestrict] at hf
state_before: R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
x : ↥((Quotient.mk I).comp C).range
hx : C x ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
⊢ x = 0
state_after: case mk
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
x : R[X] ⧸ I
hx' : x ∈ ((Quotient.mk I).comp C).range
hx : C ⟨x, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
⊢ ⟨x, hx'⟩ = 0
tactic: obtain ⟨x, hx'⟩ := x
state_before: case mk
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
x : R[X] ⧸ I
hx' : x ∈ ((Quotient.mk I).comp C).range
hx : C ⟨x, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
⊢ ⟨x, hx'⟩ = 0
state_after: case mk.intro
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
y : R
hx' : ((Quotient.mk I).comp C) y ∈ ((Quotient.mk I).comp C).range
hx : C ⟨((Quotient.mk I).comp C) y, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
⊢ ⟨((Quotient.mk I).comp C) y, hx'⟩ = 0
tactic: obtain ⟨y, rfl⟩ := RingHom.mem_range.1 hx'
state_before: case mk.intro
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
y : R
hx' : ((Quotient.mk I).comp C) y ∈ ((Quotient.mk I).comp C).range
hx : C ⟨((Quotient.mk I).comp C) y, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
⊢ ⟨((Quotient.mk I).comp C) y, hx'⟩ = 0
state_after: case mk.intro
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
y : R
hx' : ((Quotient.mk I).comp C) y ∈ ((Quotient.mk I).comp C).range
hx : C ⟨((Quotient.mk I).comp C) y, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
⊢ ↑⟨((Quotient.mk I).comp C) y, hx'⟩ = ↑0
tactic: refine Subtype.eq ?_
state_before: case mk.intro
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
y : R
hx' : ((Quotient.mk I).comp C) y ∈ ((Quotient.mk I).comp C).range
hx : C ⟨((Quotient.mk I).comp C) y, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
⊢ ↑⟨((Quotient.mk I).comp C) y, hx'⟩ = ↑0
state_after: case mk.intro
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
y : R
hx' : ((Quotient.mk I).comp C) y ∈ ((Quotient.mk I).comp C).range
hx : C ⟨((Quotient.mk I).comp C) y, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
⊢ C y ∈ I
tactic: simp only [RingHom.comp_apply, Quotient.eq_zero_iff_mem, ZeroMemClass.coe_zero]
state_before: case mk.intro
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
y : R
hx' : ((Quotient.mk I).comp C) y ∈ ((Quotient.mk I).comp C).range
hx : C ⟨((Quotient.mk I).comp C) y, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
⊢ C y ∈ I
state_after: case mk.intro
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
y : R
hx' : ((Quotient.mk I).comp C) y ∈ ((Quotient.mk I).comp C).range
hx : C ⟨((Quotient.mk I).comp C) y, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
⊢ C (i y) ∈ map (mapRingHom i) I
tactic: suffices C (i y) ∈ I.map (Polynomial.mapRingHom i) by
    obtain ⟨f, hf⟩ := mem_image_of_mem_map_of_surjective (Polynomial.mapRingHom i)
      (Polynomial.map_surjective _ (RingHom.rangeRestrict_surjective ((Quotient.mk I).comp C))) this
    refine sub_add_cancel (C y) f ▸ I.add_mem (hi' ?_ : C y - f ∈ I) hf.1
    rw [RingHom.mem_ker, RingHom.map_sub, hf.2, sub_eq_zero, coe_mapRingHom, map_C]
state_before: case mk.intro
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
y : R
hx' : ((Quotient.mk I).comp C) y ∈ ((Quotient.mk I).comp C).range
hx : C ⟨((Quotient.mk I).comp C) y, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
⊢ C (i y) ∈ map (mapRingHom i) I
state_after: no goals
tactic: exact hx