Theorem: LocalRing.local_hom_TFAE
File: Mathlib/RingTheory/LocalRing/RingHom/Basic.lean

Proof steps:
state_before: R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
state_after: case tfae_1_to_2
R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
⊢ IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)

R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
tactic: tfae_have 1 → 2
state_before: R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
state_after: case tfae_2_to_4
R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
⊢ ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)

R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
tactic: tfae_have 2 → 4
state_before: R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
state_after: case tfae_3_iff_4
R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
⊢ Ideal.map f (maximalIdeal R) ≤ maximalIdeal S ↔ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)

R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_3_iff_4 : Ideal.map f (maximalIdeal R) ≤ maximalIdeal S ↔ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
tactic: tfae_have 3 ↔ 4
state_before: R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_3_iff_4 : Ideal.map f (maximalIdeal R) ≤ maximalIdeal S ↔ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
state_after: case tfae_4_to_1
R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_3_iff_4 : Ideal.map f (maximalIdeal R) ≤ maximalIdeal S ↔ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
⊢ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S) → IsLocalRingHom f

R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_3_iff_4 : Ideal.map f (maximalIdeal R) ≤ maximalIdeal S ↔ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_4_to_1 : maximalIdeal R ≤ Ideal.comap f (maximalIdeal S) → IsLocalRingHom f
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
tactic: tfae_have 4 → 1
state_before: R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_3_iff_4 : Ideal.map f (maximalIdeal R) ≤ maximalIdeal S ↔ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_4_to_1 : maximalIdeal R ≤ Ideal.comap f (maximalIdeal S) → IsLocalRingHom f
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
state_after: case tfae_1_to_5
R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_3_iff_4 : Ideal.map f (maximalIdeal R) ≤ maximalIdeal S ↔ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_4_to_1 : maximalIdeal R ≤ Ideal.comap f (maximalIdeal S) → IsLocalRingHom f
⊢ IsLocalRingHom f → Ideal.comap f (maximalIdeal S) = maximalIdeal R

R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_3_iff_4 : Ideal.map f (maximalIdeal R) ≤ maximalIdeal S ↔ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_4_to_1 : maximalIdeal R ≤ Ideal.comap f (maximalIdeal S) → IsLocalRingHom f
tfae_1_to_5 : IsLocalRingHom f → Ideal.comap f (maximalIdeal S) = maximalIdeal R
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
tactic: tfae_have 1 → 5
state_before: R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_3_iff_4 : Ideal.map f (maximalIdeal R) ≤ maximalIdeal S ↔ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_4_to_1 : maximalIdeal R ≤ Ideal.comap f (maximalIdeal S) → IsLocalRingHom f
tfae_1_to_5 : IsLocalRingHom f → Ideal.comap f (maximalIdeal S) = maximalIdeal R
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
state_after: case tfae_5_to_4
R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_3_iff_4 : Ideal.map f (maximalIdeal R) ≤ maximalIdeal S ↔ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_4_to_1 : maximalIdeal R ≤ Ideal.comap f (maximalIdeal S) → IsLocalRingHom f
tfae_1_to_5 : IsLocalRingHom f → Ideal.comap f (maximalIdeal S) = maximalIdeal R
⊢ Ideal.comap f (maximalIdeal S) = maximalIdeal R → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)

R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_3_iff_4 : Ideal.map f (maximalIdeal R) ≤ maximalIdeal S ↔ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_4_to_1 : maximalIdeal R ≤ Ideal.comap f (maximalIdeal S) → IsLocalRingHom f
tfae_1_to_5 : IsLocalRingHom f → Ideal.comap f (maximalIdeal S) = maximalIdeal R
tfae_5_to_4 : Ideal.comap f (maximalIdeal S) = maximalIdeal R → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
tactic: tfae_have 5 → 4
state_before: R : Type u_1
S : Type u_2
T : Type u_3
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
tfae_1_to_2 : IsLocalRingHom f → ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S)
tfae_2_to_4 :
  ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S) → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_3_iff_4 : Ideal.map f (maximalIdeal R) ≤ maximalIdeal S ↔ maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
tfae_4_to_1 : maximalIdeal R ≤ Ideal.comap f (maximalIdeal S) → IsLocalRingHom f
tfae_1_to_5 : IsLocalRingHom f → Ideal.comap f (maximalIdeal S) = maximalIdeal R
tfae_5_to_4 : Ideal.comap f (maximalIdeal S) = maximalIdeal R → maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
⊢ [IsLocalRingHom f, ⇑f '' ↑(maximalIdeal R).toAddSubmonoid ⊆ ↑(maximalIdeal S),
      Ideal.map f (maximalIdeal R) ≤ maximalIdeal S, maximalIdeal R ≤ Ideal.comap f (maximalIdeal S),
      Ideal.comap f (maximalIdeal S) = maximalIdeal R].TFAE
state_after: no goals
tactic: tfae_finish