-- Mathlib/RingTheory/CotangentLocalizationAway.lean
lemma sq_ker_comp_le_ker_compLocalizationAwayAlgHom :
    ((localizationAway g (S := T)).comp P).ker ^ 2 ≤
      RingHom.ker (compLocalizationAwayAlgHom T g P)

Proof state:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
  ∀ {x : ((localizationAway g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : P.Ring
hx : x ∈ P.ker ^ 2
⊢ (algebraMap P.Ring (Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g)))) x = 0

Tactic:
rw [IsScalarTower.algebraMap_apply P.Ring (P.Ring ⧸ P.ker ^ 2) (Localization.Away _),
  Ideal.Quotient.algebraMap_eq, Ideal.Quotient.eq_zero_iff_mem.mpr hx, map_zero]

Result:
no goals

Proof state:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
  ∀ {x : ((localizationAway g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : P.Ring
hx : x ∈ P.ker ^ 2
⊢ x ∈ Ideal.comap ((localizationAway g).toComp P).toAlgHom (RingHom.ker (compLocalizationAwayAlgHom T g P))

Tactic:
simp only [Ideal.mem_comap, RingHom.mem_ker,
  compLocalizationAwayAlgHom_toAlgHom_toComp (T := T) g P x]

Result:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
  ∀ {x : ((localizationAway g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : P.Ring
hx : x ∈ P.ker ^ 2
⊢ (algebraMap P.Ring (Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g)))) x = 0

Proof state:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
  ∀ {x : ((localizationAway g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.map ((localizationAway g).toComp P).toAlgHom P.ker *
      Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
rw [Ideal.mul_le]

Result:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
  ∀ {x : ((localizationAway g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ ∀ r ∈ Ideal.map ((localizationAway g).toComp P).toAlgHom P.ker,
    ∀ s ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1},
      r * s ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)

Proof state:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
  ∀ {x : ((localizationAway g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ ∀ r ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1},
    ∀ s ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1},
      r * s ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
intro x hx y hy

Result:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
  ∀ {x : ((localizationAway g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : ((localizationAway g).comp P).Ring
hx : x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1}
y : ((localizationAway g).comp P).Ring
hy : y ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1}
⊢ x * y ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)

Proof state:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
  ∀ {x : ((localizationAway g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ ∀ r ∈ Ideal.map ((localizationAway g).toComp P).toAlgHom P.ker,
    ∀ s ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1},
      r * s ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
intro x hx y hy

Result:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
  ∀ {x : ((localizationAway g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : ((localizationAway g).comp P).Ring
hx : x ∈ Ideal.map ((localizationAway g).toComp P).toAlgHom P.ker
y : ((localizationAway g).comp P).Ring
hy : y ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1}
⊢ x * y ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)

Proof state:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
  ∀ {x : ((localizationAway g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
      Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
rw [Ideal.mul_le]

Result:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S
hsple :
  ∀ {x : ((localizationAway g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ ∀ r ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1},
    ∀ s ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1},
      r * s ∈ RingHom.ker (compLocalizationAwayAlgHom T g P)