Theorem: LocalRing.local_hom_TFAE
File: Mathlib/RingTheory/Ideal/LocalRing.lean

tactic: tfae_have 1 → 2
state_before: R : Type u
S : Type v
T : Type w
K : Type u'
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
S : Type v
T : Type w
K : Type u'
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
S : Type v
T : Type w
K : Type u'
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 2 → 4
state_before: R : Type u
S : Type v
T : Type w
K : Type u'
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
S : Type v
T : Type w
K : Type u'
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
S : Type v
T : Type w
K : Type u'
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 3 ↔ 4
state_before: R : Type u
S : Type v
T : Type w
K : Type u'
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
S : Type v
T : Type w
K : Type u'
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
S : Type v
T : Type w
K : Type u'
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 4 → 1
state_before: R : Type u
S : Type v
T : Type w
K : Type u'
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
S : Type v
T : Type w
K : Type u'
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
S : Type v
T : Type w
K : Type u'
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 1 → 5
state_before: R : Type u
S : Type v
T : Type w
K : Type u'
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
S : Type v
T : Type w
K : Type u'
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
S : Type v
T : Type w
K : Type u'
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 5 → 4
state_before: R : Type u
S : Type v
T : Type w
K : Type u'
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
S : Type v
T : Type w
K : Type u'
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
S : Type v
T : Type w
K : Type u'
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_finish
state_before: R : Type u
S : Type v
T : Type w
K : Type u'
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: rintro _ _ ⟨a, ha, rfl⟩
state_before: case tfae_1_to_2
R : Type u
S : Type v
T : Type w
K : Type u'
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)
state_after: case tfae_1_to_2.intro.intro
R : Type u
S : Type v
T : Type w
K : Type u'
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
a✝ : IsLocalRingHom f
a : R
ha : a ∈ ↑(maximalIdeal R).toAddSubmonoid
⊢ f a ∈ ↑(maximalIdeal S)

tactic: exact map_nonunit f a ha
state_before: case tfae_1_to_2.intro.intro
R : Type u
S : Type v
T : Type w
K : Type u'
inst✝³ : CommSemiring R
inst✝² : LocalRing R
inst✝¹ : CommSemiring S
inst✝ : LocalRing S
f : R →+* S
a✝ : IsLocalRingHom f
a : R
ha : a ∈ ↑(maximalIdeal R).toAddSubmonoid
⊢ f a ∈ ↑(maximalIdeal S)
state_after: no goals

tactic: exact Set.image_subset_iff.1
state_before: case tfae_2_to_4
R : Type u
S : Type v
T : Type w
K : Type u'
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)
state_after: no goals

tactic: exact Ideal.map_le_iff_le_comap
state_before: case tfae_3_iff_4
R : Type u
S : Type v
T : Type w
K : Type u'
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)
state_after: no goals

tactic: intro h
state_before: case tfae_4_to_1
R : Type u
S : Type v
T : Type w
K : Type u'
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
state_after: case tfae_4_to_1
R : Type u
S : Type v
T : Type w
K : Type u'
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)
h : maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
⊢ IsLocalRingHom f

tactic: constructor
state_before: case tfae_4_to_1
R : Type u
S : Type v
T : Type w
K : Type u'
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)
h : maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
⊢ IsLocalRingHom f
state_after: case tfae_4_to_1.map_nonunit
R : Type u
S : Type v
T : Type w
K : Type u'
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)
h : maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
⊢ ∀ (a : R), IsUnit (f a) → IsUnit a

tactic: exact fun x => not_imp_not.1 (@h x)
state_before: case tfae_4_to_1.map_nonunit
R : Type u
S : Type v
T : Type w
K : Type u'
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)
h : maximalIdeal R ≤ Ideal.comap f (maximalIdeal S)
⊢ ∀ (a : R), IsUnit (f a) → IsUnit a
state_after: no goals

tactic: intro
state_before: case tfae_1_to_5
R : Type u
S : Type v
T : Type w
K : Type u'
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
state_after: case tfae_1_to_5
R : Type u
S : Type v
T : Type w
K : Type u'
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
a✝ : IsLocalRingHom f
⊢ Ideal.comap f (maximalIdeal S) = maximalIdeal R

tactic: ext
state_before: case tfae_1_to_5
R : Type u
S : Type v
T : Type w
K : Type u'
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
a✝ : IsLocalRingHom f
⊢ Ideal.comap f (maximalIdeal S) = maximalIdeal R
state_after: case tfae_1_to_5.h
R : Type u
S : Type v
T : Type w
K : Type u'
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
a✝ : IsLocalRingHom f
x✝ : R
⊢ x✝ ∈ Ideal.comap f (maximalIdeal S) ↔ x✝ ∈ maximalIdeal R

tactic: exact not_iff_not.2 (isUnit_map_iff f _)
state_before: case tfae_1_to_5.h
R : Type u
S : Type v
T : Type w
K : Type u'
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
a✝ : IsLocalRingHom f
x✝ : R
⊢ x✝ ∈ Ideal.comap f (maximalIdeal S) ↔ x✝ ∈ maximalIdeal R
state_after: no goals

tactic: exact fun h => le_of_eq h.symm
state_before: case tfae_5_to_4
R : Type u
S : Type v
T : Type w
K : Type u'
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)
state_after: no goals