-- Module: Mathlib.Topology.Algebra.Valued.ValuedField
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
s : Set (hat K)
this :
  HasBasis.{u_1, u_2 + 1} (α := hat K) (𝓝 0) (fun (x : (ValueGroup₀ (F := Valuation K Γ₀) v)ˣ) ↦ True)
    fun (γ : (ValueGroup₀ (F := Valuation K Γ₀) v)ˣ) ↦
    {x : hat K |
      LT.lt.{u_2} (α := Γ₀) ((extensionValuation : (a : hat K) → Γ₀) x : Γ₀)
        ((↑((Units.map (↑embedding : ValueGroup₀ v →* Γ₀) : (a : (ValueGroup₀ v)ˣ) → Γ₀ˣ) γ) : Γ₀) : Γ₀)}
x : hat K
γ : (ValueGroup₀ (F := Valuation K Γ₀) v)ˣ
hx0 : ¬x = 0
y : K :=
  Exists.choose (p := fun (a : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) a =
      (↑γ : ValueGroup₀ v))
    ⋯
hy_def :
  y =
    Exists.choose (p := fun (a : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a =
        (↑γ : ValueGroup₀ v))
      ⋯
hy : ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
    (Exists.choose (p := fun (a : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a =
        (↑γ : ValueGroup₀ v))
      ⋯) =
  (↑γ : ValueGroup₀ v) :=
  ⋯
neg :
  LT.lt (α := ValueGroup₀ (F := MonoidWithZeroHom.{u_1, u_2} (hat K) Γ₀) (↑extensionValuation : hat K →*₀ Γ₀))
      ((↑(Subtype.mk (p := fun (x : Γ₀ˣ) ↦
              Membership.mem (γ := Subgroup Γ₀ˣ) (valueGroup (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
                x)
              (Units.mk0 ((embedding : (a : ValueGroup₀ v) → Γ₀) (extension x)) ⋯) ⋯) :
          WithZero ↥(valueGroup (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)) :
        WithZero ↥(valueGroup (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation))
      ((Valuation.restrict extensionValuation : (a : hat K) → ValueGroup₀ (↑extensionValuation : hat K →*₀ Γ₀))
          (↑(⋯.choose (p := fun (a : K) ↦
                ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) :
                      (a : K) → ValueGroup₀ v)
                    a =
                  (↑γ : ValueGroup₀ v))) :
            hat K) :
        ValueGroup₀ (F := MonoidWithZeroHom.{u_1, u_2} (hat K) Γ₀) (↑extensionValuation : hat K →*₀ Γ₀)) ↔
    extension x < (↑γ : ValueGroup₀ v)
⊢ LT.lt.{u_2} (α := ValueGroup₀ (F := MonoidWithZeroHom.{u_1, u_2} (hat K) Γ₀) (↑extensionValuation : hat K →*₀ Γ₀))
      ((↑(Subtype.mk (p := fun (x : Γ₀ˣ) ↦
              Membership.mem (γ := Subgroup Γ₀ˣ) (valueGroup (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
                x)
              (Units.mk0 ((extensionValuation : (a : hat K) → Γ₀) x) ⋯) ⋯) :
          WithZero ↥(valueGroup (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)) :
        WithZero ↥(valueGroup (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation))
      ((valueGroup₀_equiv_extensionValuation : (a : ValueGroup₀ v) → ValueGroup₀ extensionValuation)
          (↑γ : ValueGroup₀ v) :
        ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation) ↔
    extension x < (↑γ : ValueGroup₀ v)

Tactic:
simp only [extensionValuation_toFun, valueGroup₀_equiv_extensionValuation,
              valueGroup₀_hom_extensionValuation,
              MulEquiv.ofBijective_apply, coe_mk, ZeroHom.coe_mk]

Premises used: MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, Valued.valueGroup₀_equiv_extensionValuation, MulEquiv.ofBijective_apply, Valued.valueGroup₀_hom_extensionValuation, Valued.extensionValuation_toFun