-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
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 :=
  ⋯
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)
    (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)
      ⋯) =
  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)))
have rw : ∀
  (hx :
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) x =
      a),
  let 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)
      sorry;
  ∀
    (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)
          sorry),
    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)
        sorry;
    ∀
      (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)
            sorry),
      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) sorry) sorry) :
            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) sorry) sorry) :
                  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) sorry) sorry) :
                  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: rw [← hx_def] at hx
Cleared hypotheses: hx, y, hy_def, hy, xy, hxy_def, hxy