Theorem: idealFactorsFunOfQuotHom_coe_coe
File: Mathlib/RingTheory/DedekindDomain/Ideal.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : { p // p ∣ I }
⊢ comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)) ∣ J
state_after: R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : { p // p ∣ I }
this : RingHom.ker (Ideal.Quotient.mk J) ≤ comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X))
⊢ comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)) ∣ J
tactic: have : RingHom.ker (Ideal.Quotient.mk J) ≤
        comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) X)) :=
      ker_le_comap (Ideal.Quotient.mk J)
state_before: R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : { p // p ∣ I }
this : RingHom.ker (Ideal.Quotient.mk J) ≤ comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X))
⊢ comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)) ∣ J
state_after: R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : { p // p ∣ I }
this : J ≤ comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X))
⊢ comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)) ∣ J
tactic: rw [mk_ker] at this
state_before: R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : { p // p ∣ I }
this : J ≤ comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X))
⊢ comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)) ∣ J
state_after: no goals
tactic: exact dvd_iff_le.mpr this
state_before: R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
⊢ Monotone fun X => ⟨comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)), ⋯⟩
state_after: case mk.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : Ideal R
hX : X ∣ I
Y : Ideal R
hY : Y ∣ I
h : ⟨X, hX⟩ ≤ ⟨Y, hY⟩
⊢ (fun X => ⟨comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)), ⋯⟩) ⟨X, hX⟩ ≤
    (fun X => ⟨comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)), ⋯⟩) ⟨Y, hY⟩
tactic: rintro ⟨X, hX⟩ ⟨Y, hY⟩ h
state_before: case mk.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : Ideal R
hX : X ∣ I
Y : Ideal R
hY : Y ∣ I
h : ⟨X, hX⟩ ≤ ⟨Y, hY⟩
⊢ (fun X => ⟨comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)), ⋯⟩) ⟨X, hX⟩ ≤
    (fun X => ⟨comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)), ⋯⟩) ⟨Y, hY⟩
state_after: case mk.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : Ideal R
hX : X ∣ I
Y : Ideal R
hY : Y ∣ I
h : X ≤ Y
⊢ comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) X)) ≤
    ↑((fun X => ⟨comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)), ⋯⟩) ⟨Y, hY⟩)
tactic: rw [← Subtype.coe_le_coe, Subtype.coe_mk, Subtype.coe_mk] at h ⊢
state_before: case mk.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : Ideal R
hX : X ∣ I
Y : Ideal R
hY : Y ∣ I
h : X ≤ Y
⊢ comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) X)) ≤
    ↑((fun X => ⟨comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)), ⋯⟩) ⟨Y, hY⟩)
state_after: case mk.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : Ideal R
hX : X ∣ I
Y : Ideal R
hY : Y ∣ I
h : X ≤ Y
⊢ map (Ideal.Quotient.mk I) X ≤ map (Ideal.Quotient.mk I) Y ⊔ comap f ⊥
tactic: rw [Subtype.coe_mk, comap_le_comap_iff_of_surjective (Ideal.Quotient.mk J)
      Quotient.mk_surjective, map_le_iff_le_comap, Subtype.coe_mk, comap_map_of_surjective _ hf
      (map (Ideal.Quotient.mk I) Y)]
state_before: case mk.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : Ideal R
hX : X ∣ I
Y : Ideal R
hY : Y ∣ I
h : X ≤ Y
⊢ map (Ideal.Quotient.mk I) X ≤ map (Ideal.Quotient.mk I) Y ⊔ comap f ⊥
state_after: case mk.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : Ideal R
hX : X ∣ I
Y : Ideal R
hY : Y ∣ I
h : X ≤ Y
⊢ map (Ideal.Quotient.mk I) X ≤ map (Ideal.Quotient.mk I) Y
tactic: suffices map (Ideal.Quotient.mk I) X ≤ map (Ideal.Quotient.mk I) Y by
      exact le_sup_of_le_left this
state_before: case mk.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain A
I : Ideal R
J : Ideal A
f : R ⧸ I →+* A ⧸ J
hf : Function.Surjective ⇑f
X : Ideal R
hX : X ∣ I
Y : Ideal R
hY : Y ∣ I
h : X ≤ Y
⊢ map (Ideal.Quotient.mk I) X ≤ map (Ideal.Quotient.mk I) Y
state_after: no goals
tactic: rwa [map_le_iff_le_comap, comap_map_of_surjective (Ideal.Quotient.mk I)
      Quotient.mk_surjective, ← RingHom.ker_eq_comap_bot, mk_ker, sup_eq_left.mpr <| le_of_dvd hY]