Module: Mathlib.NumberTheory.RamificationInertia.Valuation

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁶ : CommRing A
inst✝¹⁵ : IsDedekindDomain A
inst✝¹⁴ : CommRing B
inst✝¹³ : IsDedekindDomain B
inst✝¹² : Algebra A B
inst✝¹¹ : Module.IsTorsionFree A B
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
inst✝⁷ : Algebra A K
inst✝⁶ : IsFractionRing A K
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A K L
inst✝³ : Algebra B L
inst✝² : IsFractionRing B L
inst✝¹ : IsScalarTower A B L
v : HeightOneSpectrum A
w : HeightOneSpectrum B
inst✝ : w.asIdeal.LiesOver v.asIdeal
γL : (ValuativeRel.ValueGroupWithZero (WithVal (valuation L w)))ˣ
a✝ : True
e : ℕ := v.asIdeal.ramificationIdx' w.asIdeal
σL : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ ≃*o (MonoidWithZeroHom.ofClass (valuation L w)).ValueGroup₀ :=
  WithVal.valueGroupOrderIso₀ (valuation L w)
σw : (MonoidWithZeroHom.ofClass (valuation L w)).ValueGroup₀ ≃*o WithZero (Multiplicative ℤ) :=
  valueGroup₀_equiv_withZeroMulInt (valuation L w)
σwV : ValuativeRel.ValueGroupWithZero (WithVal (valuation L w)) ≃*o
  (MonoidWithZeroHom.ofClass (WithVal.valuation (valuation L w))).ValueGroup₀ :=
  ValuativeRel.ValueGroupWithZero.orderMonoidIso (WithVal.valuation (valuation L w))
m : WithZero (Multiplicative ℤ) := σw (σL (σwV ↑γL))
σvV : ValuativeRel.ValueGroupWithZero (WithVal (valuation K v)) ≃*o
  (MonoidWithZeroHom.ofClass (WithVal.valuation (valuation K v))).ValueGroup₀ :=
  ValuativeRel.ValueGroupWithZero.orderMonoidIso (WithVal.valuation (valuation K v))
σv : (MonoidWithZeroHom.ofClass (valuation K v)).ValueGroup₀ ≃*o WithZero (Multiplicative ℤ) :=
  valueGroup₀_equiv_withZeroMulInt (valuation K v)
⊢ ∃ ia,
    True ∧
      ∀ x ∈ {x | (ValuativeRel.valuation (WithVal (valuation K v))) x < ↑ia},
        (algebraMap (WithVal (valuation K v)) (WithVal (valuation L w))) x ∈
          {x | (ValuativeRel.valuation (WithVal (valuation L w))) x < ↑γL}

Tactic:
let σK := WithVal.valueGroupOrderIso₀ (v.valuation K)

Goal state after:
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁶ : CommRing A
inst✝¹⁵ : IsDedekindDomain A
inst✝¹⁴ : CommRing B
inst✝¹³ : IsDedekindDomain B
inst✝¹² : Algebra A B
inst✝¹¹ : Module.IsTorsionFree A B
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
inst✝⁷ : Algebra A K
inst✝⁶ : IsFractionRing A K
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A K L
inst✝³ : Algebra B L
inst✝² : IsFractionRing B L
inst✝¹ : IsScalarTower A B L
v : HeightOneSpectrum A
w : HeightOneSpectrum B
inst✝ : w.asIdeal.LiesOver v.asIdeal
γL : (ValuativeRel.ValueGroupWithZero (WithVal (valuation L w)))ˣ
a✝ : True
e : ℕ := v.asIdeal.ramificationIdx' w.asIdeal
σL : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ ≃*o (MonoidWithZeroHom.ofClass (valuation L w)).ValueGroup₀ :=
  WithVal.valueGroupOrderIso₀ (valuation L w)
σw : (MonoidWithZeroHom.ofClass (valuation L w)).ValueGroup₀ ≃*o WithZero (Multiplicative ℤ) :=
  valueGroup₀_equiv_withZeroMulInt (valuation L w)
σwV : ValuativeRel.ValueGroupWithZero (WithVal (valuation L w)) ≃*o
  (MonoidWithZeroHom.ofClass (WithVal.valuation (valuation L w))).ValueGroup₀ :=
  ValuativeRel.ValueGroupWithZero.orderMonoidIso (WithVal.valuation (valuation L w))
m : WithZero (Multiplicative ℤ) := σw (σL (σwV ↑γL))
σvV : ValuativeRel.ValueGroupWithZero (WithVal (valuation K v)) ≃*o
  (MonoidWithZeroHom.ofClass (WithVal.valuation (valuation K v))).ValueGroup₀ :=
  ValuativeRel.ValueGroupWithZero.orderMonoidIso (WithVal.valuation (valuation K v))
σv : (MonoidWithZeroHom.ofClass (valuation K v)).ValueGroup₀ ≃*o WithZero (Multiplicative ℤ) :=
  valueGroup₀_equiv_withZeroMulInt (valuation K v)
σK : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ ≃*o (MonoidWithZeroHom.ofClass (valuation K v)).ValueGroup₀ :=
  WithVal.valueGroupOrderIso₀ (valuation K v)
⊢ ∃ ia,
    True ∧
      ∀ x ∈ {x | (ValuativeRel.valuation (WithVal (valuation K v))) x < ↑ia},
        (algebraMap (WithVal (valuation K v)) (WithVal (valuation L w))) x ∈
          {x | (ValuativeRel.valuation (WithVal (valuation L w))) x < ↑γL}