Module: Mathlib.Analysis.Normed.Unbundled.FiniteExtension

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : NormedField K
inst✝¹ : Field L
inst✝ : Algebra K L
hfd : FiniteDimensional K L
hna : IsNonarchimedean norm
⊢ ∃ f, IsPowMul ⇑f ∧ (∀ (x : K), f ((algebraMap K L) x) = ‖x‖) ∧ IsNonarchimedean ⇑f

Tactic:
by
  -- Choose a basis B = {1, e2,..., en} of the K-vector space L
  have h1 : LinearIndepOn K id ({1} : Set L) := .singleton one_ne_zero
  set ι := { x // x ∈ LinearIndepOn.extend h1 (Set.subset_univ ({1} : Set L)) }
  set B : Basis ι K L := Basis.extend h1
  let hfin : Fintype ι := FiniteDimensional.fintypeBasisIndex B
  have hem : Nonempty ι := B.index_nonempty
  have h1L : (1 : L) ∈ LinearIndepOn.extend h1 _ :=
    Basis.subset_extend _ (Set.mem_singleton (1 : L))
  have hB1 : B ⟨1, h1L⟩ = (1 : L) := by rw [Basis.coe_extend, Subtype.coe_mk]
  -- Define a function g : L → ℝ by setting g (∑ki • ei) = maxᵢ ‖ ki ‖
  set g : L → ℝ := B.norm
  -- g 0 = 0seminormFromBounded
  have hg0 : g 0 = 0 := B.norm_zero
  -- g takes nonnegative values
  have hg_nonneg : ∀ x : L, 0 ≤ g x := fun x ↦ by simp only [g, Basis.norm]; simp
  -- g extends the norm on K
  have hg_ext : ∀ (x : K), g ((algebraMap K L) x) = ‖x‖ := Basis.norm_extends hB1
  -- g is nonarchimedean
  have hg_na : IsNonarchimedean g := Basis.norm_isNonarchimedean hna
  -- g satisfies the triangle inequality
  have hg_add : ∀ a b : L, g (a + b) ≤ g a + g b :=
    fun _ _ ↦ IsNonarchimedean.add_le hg_nonneg hg_na
  -- g (-a) = g a
  have hg_neg : ∀ a : L, g (-a) = g a := B.norm_neg
  -- g is multiplicatively bounded
  obtain ⟨_, _, hg_bdd⟩ := Basis.norm_mul_le_const_mul_norm hB1 hna
  -- g is a K-module norm
  have hg_mul : ∀ (k : K) (y : L), g ((algebraMap K L) k * y) = g ((algebraMap K L) k) * g y :=
    fun k y ↦ Basis.norm_smul hB1 k y
  -- Using BGR Prop. 1.2.1/2, we can smooth g to a ring norm f on L that extends the norm on K.
  set f := seminormFromBounded hg0 hg_nonneg hg_bdd hg_add hg_neg
  have hf_na : IsNonarchimedean f := seminormFromBounded_isNonarchimedean hg_nonneg hg_bdd hg_na
  have hf_1 : f 1 ≤ 1 := seminormFromBounded_one_le hg_nonneg hg_bdd
  have hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖ :=
    fun k ↦ hg_ext k ▸ seminormFromBounded_of_mul_apply hg_nonneg hg_bdd (hg_mul k)
  -- Using BGR Prop. 1.3.2/1, we obtain from f a power multiplicative K-algebra norm on L
  -- extending the norm on K.
  set F' := smoothingSeminorm f hf_1 hf_na with hF'
  have hF'_ext : ∀ k : K, F' ((algebraMap K L) k) = ‖k‖ := by
    intro k
    rw [← hf_ext _]
    exact smoothingSeminorm_apply_of_map_mul_eq_mul f hf_1 hf_na
      (seminormFromBounded_of_mul_is_mul hg_nonneg hg_bdd (hg_mul k))
  have hF'_1 : F' 1 = 1 := by
    have h1 : (1 : L) = (algebraMap K L) 1 := by rw [map_one]
    simp only [h1, hF'_ext (1 : K), norm_one]
  have hF'_0 : F' ≠ 0 := DFunLike.ne_iff.mpr ⟨(1 : L), by rw [hF'_1]; exact one_ne_zero⟩
  set F : AlgebraNorm K L :=
    { RingSeminorm.toRingNorm F' hF'_0 with
      smul' := fun k y ↦ by
        have hk : ∀ y : L, f (algebraMap K L k * y) = f (algebraMap K L k) * f y :=
          seminormFromBounded_of_mul_is_mul hg_nonneg hg_bdd (hg_mul k)
        have hfk : ‖k‖ = (smoothingSeminorm f hf_1 hf_na) ((algebraMap K L) k) := by
          rw [← hf_ext k, eq_comm, smoothingSeminorm_apply_of_map_mul_eq_mul f hf_1 hf_na hk]
        simp only [hfk, hF']
        -- TODO: There are missing `simp` lemmas here, that should be able to convert
        -- `((smoothingSeminorm f hf_1 hf_na).toRingNorm ⋯).toRingSeminorm y` to
        -- `(smoothingSeminorm f hf_1 hf_na y)`, after which the `erw` would work as a `rw`.
        erw [← smoothingSeminorm_of_mul f hf_1 hf_na hk y]
        rw [Algebra.smul_def]
        rfl }
  have hF_ext (k : K) : F ((algebraMap K L) k) = ‖k‖ := by
    rw [← hf_ext]
    exact smoothingSeminorm_apply_of_map_mul_eq_mul f hf_1 hf_na
      (seminormFromBounded_of_mul_is_mul hg_nonneg hg_bdd (hg_mul k))
  exact ⟨F, isPowMul_smoothingFun f hf_1, hF_ext, isNonarchimedean_smoothingFun f hf_1 hf_na⟩

Goal state after: