theorem continuous_extension : Continuous (Valued.extension : hat K → Γ₀) := by
  refine' Completion.denseInducing_coe.continuous_extend _
  intro x₀
  rcases eq_or_ne x₀ 0 with (rfl | h)
  · refine' ⟨0, _⟩
    erw [← Completion.denseInducing_coe.toInducing.nhds_eq_comap]
    exact Valued.continuous_valuation.tendsto' 0 0 (map_zero v)
  · have preimage_one : v ⁻¹' {(1 : Γ₀)} ∈ 𝓝 (1 : K) := by
      have : (v (1 : K) : Γ₀) ≠ 0 := by
        rw [Valuation.map_one]
        exact zero_ne_one.symm
      convert Valued.loc_const this
      ext x
      rw [Valuation.map_one, mem_preimage, mem_singleton_iff, mem_setOf_eq]
    obtain ⟨V, V_in, hV⟩ : ∃ V ∈ 𝓝 (1 : hat K), ∀ x : K, (x : hat K) ∈ V → (v x : Γ₀) = 1 := by
      rwa [Completion.denseInducing_coe.nhds_eq_comap, mem_comap] at preimage_one
    have : ∃ V' ∈ 𝓝 (1 : hat K), (0 : hat K) ∉ V' ∧ ∀ (x) (_ : x ∈ V') (y) (_ : y ∈ V'),
      x * y⁻¹ ∈ V := by
      have : Tendsto (fun p : hat K × hat K => p.1 * p.2⁻¹) ((𝓝 1) ×ˢ (𝓝 1)) (𝓝 1) := by
        rw [← nhds_prod_eq]
        conv =>
          congr
          rfl
          rfl
          rw [← one_mul (1 : hat K)]
        refine'
          Tendsto.mul continuous_fst.continuousAt (Tendsto.comp _ continuous_snd.continuousAt)
        -- Porting note: Added `ContinuousAt.tendsto`
        convert (continuousAt_inv₀ (zero_ne_one.symm : 1 ≠ (0 : hat K))).tendsto
        exact inv_one.symm
      rcases tendsto_prod_self_iff.mp this V V_in with ⟨U, U_in, hU⟩
      let hatKstar := ({0}ᶜ : Set <| hat K)
      have : hatKstar ∈ 𝓝 (1 : hat K) := compl_singleton_mem_nhds zero_ne_one.symm
      use U ∩ hatKstar, Filter.inter_mem U_in this
      constructor
      · rintro ⟨_, h'⟩
        rw [mem_compl_singleton_iff] at h'
        exact h' rfl
      · rintro x ⟨hx, _⟩ y ⟨hy, _⟩
        apply hU <;> assumption
    rcases this with ⟨V', V'_in, zeroV', hV'⟩
    have nhds_right : (fun x => x * x₀) '' V' ∈ 𝓝 x₀ := by
      have l : Function.LeftInverse (fun x : hat K => x * x₀⁻¹) fun x : hat K => x * x₀ := by
        intro x
        simp only [mul_assoc, mul_inv_cancel h, mul_one]
      have r : Function.RightInverse (fun x : hat K => x * x₀⁻¹) fun x : hat K => x * x₀ := by
        intro x
        simp only [mul_assoc, inv_mul_cancel h, mul_one]
      have c : Continuous fun x : hat K => x * x₀⁻¹ := continuous_id.mul continuous_const
      rw [image_eq_preimage_of_inverse l r]
      rw [← mul_inv_cancel h] at V'_in
      exact c.continuousAt V'_in
    have : ∃ z₀ : K, ∃ y₀ ∈ V', ↑z₀ = y₀ * x₀ ∧ z₀ ≠ 0 := by
      rcases Completion.denseRange_coe.mem_nhds nhds_right with ⟨z₀, y₀, y₀_in, H : y₀ * x₀ = z₀⟩
      refine' ⟨z₀, y₀, y₀_in, ⟨H.symm, _⟩⟩
      rintro rfl
      exact mul_ne_zero (ne_of_mem_of_not_mem y₀_in zeroV') h H
    rcases this with ⟨z₀, y₀, y₀_in, hz₀, z₀_ne⟩
    have vz₀_ne : (v z₀ : Γ₀) ≠ 0 := by rwa [Valuation.ne_zero_iff]
    refine' ⟨v z₀, _⟩
    rw [WithZeroTopology.tendsto_of_ne_zero vz₀_ne, eventually_comap]
    filter_upwards [nhds_right] with x x_in a ha
    rcases x_in with ⟨y, y_in, rfl⟩
    have : (v (a * z₀⁻¹) : Γ₀) = 1 := by
      apply hV
      have : (z₀⁻¹ : K) = (z₀ : hat K)⁻¹ := map_inv₀ (Completion.coeRingHom : K →+* hat K) z₀
      rw [Completion.coe_mul, this, ha, hz₀, mul_inv, mul_comm y₀⁻¹, ← mul_assoc, mul_assoc y,
        mul_inv_cancel h, mul_one]
      solve_by_elim
    

Proof state:
case h.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑K x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x => x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑K z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne : v z₀ ≠ 0
a : K
y : hat K
y_in : y ∈ V'
ha : ↑K a = (fun x => x * x₀) y
this : v (a * z₀⁻¹) = 1
⊢ v a = v z₀

Next tactic:
calc
      v a = v (a * z₀⁻¹ * z₀) := by rw [mul_assoc, inv_mul_cancel z₀_ne, mul_one]
      _ = v (a * z₀⁻¹) * v z₀ := (Valuation.map_mul _ _ _)
      _ = v z₀ := by rw [this, one_mul]