-- 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
hab :
  Eq.{u_2 + 1} (α := ValueGroup₀ (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_1 : K) ↦
              ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) :
                    (a : K) → ValueGroup₀ v)
                  a_1 =
                a)) :
          hat K) :
      ValueGroup₀ (F := MonoidWithZeroHom.{u_1, u_2} (hat K) Γ₀) (↑extensionValuation : hat K →*₀ Γ₀))
    ((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 =
                b)) :
          hat K) :
      ValueGroup₀ (F := MonoidWithZeroHom.{u_1, u_2} (hat K) Γ₀) (↑extensionValuation : hat K →*₀ Γ₀))
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 :=
  ⋯
this :
  Eq.{u_2 + 1} (α := ValueGroup₀ (F := MonoidWithZeroHom.{u_1, u_2} (hat K) Γ₀) (↑extensionValuation : hat K →*₀ Γ₀))
    (if h :
        Eq (α := ValueGroup₀ (F := Valuation K Γ₀) v) (extension ((algebraMap K (hat K) : (a : K) → hat K) x)) 0 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 ((embedding : (a : ValueGroup₀ v) → Γ₀) (extension ((algebraMap K (hat K) : (a : K) → hat K) x)))
              ⋯)
            ⋯) :
        WithZero ↥(valueGroup extensionValuation)))
    (if h :
        Eq (α := ValueGroup₀ (F := Valuation K Γ₀) v) (extension ((algebraMap K (hat K) : (a : K) → hat K) y)) 0 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 ((embedding : (a : ValueGroup₀ v) → Γ₀) (extension ((algebraMap K (hat K) : (a : K) → hat K) y)))
              ⋯)
            ⋯) :
        WithZero ↥(valueGroup extensionValuation)))
ha0 : ¬a = 0
h0 : Ne (α := ValueGroup₀ (F := Valuation K Γ₀) v) (extension ((algebraMap K (hat K) : (a : K) → hat K) x)) 0
hx_def₁ :
  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)
      ⋯
⊢ a = b

Tactic:
rw [← hx_def, ← hy_def, Valuation.restrict_inj, extensionValuation_toFun,
            extensionValuation_toFun, embedding_strictMono.injective.eq_iff,
            extension_extends, extension_extends, v.restrict_def, v.restrict_def,
            hx, hy] at hab

Premises used: Valued.extension_extends, Valued.v, Function.Injective.eq_iff, Valuation.restrict_inj, Valuation.restrict_def, StrictMono.injective, Valued.extensionValuation_toFun