-- 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)
    (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)
      ⋯) =
  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)
    (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)
      ⋯) =
  b :=
  ⋯
set :
  let 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_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)
          ⋯),
    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)
              (↑(Exists.choose (p := fun (a_1 : K) ↦
                    (if h : Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) a_1 : Γ₀) (0 : Γ₀) then 0
                      else
                        (↑(Subtype.mk (p := fun (x : Γ₀ˣ) ↦
                              Membership.mem (γ := Subgroup Γ₀ˣ) (valueGroup (F := Valuation K Γ₀) v) x)
                              (Units.mk0 ((v : (a : K) → Γ₀) a_1) h) ⋯) :
                          WithZero ↥(valueGroup v))) =
                      a * b)
                    ⋯) :
                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)
                (Exists.choose (p := fun (a_1 : K) ↦
                  (if h : Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) a_1 : Γ₀) (0 : Γ₀) then 0
                    else
                      (↑(Subtype.mk (p := fun (x : Γ₀ˣ) ↦
                            Membership.mem (γ := Subgroup Γ₀ˣ) (valueGroup (F := Valuation K Γ₀) v) x)
                            (Units.mk0 ((v : (a : K) → Γ₀) a_1) h) ⋯) :
                        WithZero ↥(valueGroup v))) =
                    a * b)
                  ⋯))
              ⋯)
            ⋯) :
        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:
set xy := (restrict₀_surjective hv.v (a * b)).choose with hxy_def

Premises used: HMul.hMul, Valued.v, MonoidWithZeroHom.ValueGroup₀.restrict₀_surjective, Exists.choose