-- 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
a b : ValueGroup₀ (F := Valuation K Γ₀) v
x : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a)
    ⋯
hx_def :
  x =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a)
      ⋯
hx :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) x = a
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 =
      b)
    ⋯
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 =
        b)
      ⋯
hy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) y = b
xy : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a * b)
    ⋯
hxy_def :
  xy =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a * b)
      ⋯
hxy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) xy =
    a * b
hx0 : ¬x = 0
pos :
  ∀ (hy0 : y = 0),
    Eq.{u_2 + 1} (α := ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
      (if h : Eq.{u_2 + 1} (α := Γ₀) ((extensionValuation : (a : hat K) → Γ₀) (↑xy : hat K) : Γ₀) (0 : Γ₀) then 0
      else
        (↑(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 ((v : (a : K) → Γ₀) xy) ⋯) ⋯) :
          WithZero ↥(valueGroup extensionValuation)))
      (if h : Eq.{u_2 + 1} (α := Γ₀) ((extensionValuation : (a : hat K) → Γ₀) (↑y : hat K) : Γ₀) (0 : Γ₀) then 0
      else
        if h : Eq.{u_2 + 1} (α := Γ₀) ((extensionValuation : (a : hat K) → Γ₀) (↑x : hat K) : Γ₀) (0 : Γ₀) then 0
        else
          HMul.hMul (α := ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation) (β :=
            ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
            ((↑(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 ((v : (a : 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))
            ((↑(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 ((v : (a : K) → Γ₀) y) ⋯) ⋯) :
                WithZero ↥(valueGroup (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)) :
              WithZero ↥(valueGroup (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)))
neg :
  ∀ (hy0 : ¬y = 0),
    Eq.{u_2 + 1} (α := ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
      (if h : Eq.{u_2 + 1} (α := Γ₀) ((extensionValuation : (a : hat K) → Γ₀) (↑xy : hat K) : Γ₀) (0 : Γ₀) then 0
      else
        (↑(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 ((v : (a : K) → Γ₀) xy) ⋯) ⋯) :
          WithZero ↥(valueGroup extensionValuation)))
      (if h : Eq.{u_2 + 1} (α := Γ₀) ((extensionValuation : (a : hat K) → Γ₀) (↑y : hat K) : Γ₀) (0 : Γ₀) then 0
      else
        if h : Eq.{u_2 + 1} (α := Γ₀) ((extensionValuation : (a : hat K) → Γ₀) (↑x : hat K) : Γ₀) (0 : Γ₀) then 0
        else
          HMul.hMul (α := ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation) (β :=
            ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
            ((↑(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 ((v : (a : 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))
            ((↑(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 ((v : (a : K) → Γ₀) y) ⋯) ⋯) :
                WithZero ↥(valueGroup (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)) :
              WithZero ↥(valueGroup (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)))
⊢ Eq.{u_2 + 1} (α := ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
    (if h : Eq.{u_2 + 1} (α := Γ₀) ((extensionValuation : (a : hat K) → Γ₀) (↑xy : hat K) : Γ₀) (0 : Γ₀) then 0
    else
      (↑(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 ((v : (a : K) → Γ₀) xy) ⋯) ⋯) :
        WithZero ↥(valueGroup extensionValuation)))
    (if h : Eq.{u_2 + 1} (α := Γ₀) ((extensionValuation : (a : hat K) → Γ₀) (↑y : hat K) : Γ₀) (0 : Γ₀) then 0
    else
      if h : Eq.{u_2 + 1} (α := Γ₀) ((extensionValuation : (a : hat K) → Γ₀) (↑x : hat K) : Γ₀) (0 : Γ₀) then 0
      else
        HMul.hMul (α := ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation) (β :=
          ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
          ((↑(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 ((v : (a : 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))
          ((↑(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 ((v : (a : K) → Γ₀) y) ⋯) ⋯) :
              WithZero ↥(valueGroup (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)) :
            WithZero ↥(valueGroup (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)))

Tactic:
by_cases hy0 : y = 0