-- Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (R := 𝓞 K) (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < ((volume : Measure (mixedSpace K)) : (a : Set (mixedSpace K)) → ℝ≥0∞) (convexBodyLT K f)
h_fund :
  IsAddFundamentalDomain.{u_1, u_1}
    ↥(toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
        (span ℤ
          (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
            (⇑(fractionalIdealLatticeBasis K I :
                  Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                    (mixedSpace K)) :
              (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) → mixedSpace K))))
    (α := mixedSpace K)
    (fundamentalDomain (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) (K := ℝ)
      (fractionalIdealLatticeBasis K I))
    volume
this :
  Countable.{u_1 + 1}
    ↥(toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
        (span ℤ
          (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
            (⇑(fractionalIdealLatticeBasis K I :
                  Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                    (mixedSpace K)) :
              (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) → mixedSpace K))))
a : K
ha : a ∈ (↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Set K)
hx :
  Membership.mem (α := mixedSpace K) (γ := AddSubgroup (mixedSpace K))
    (toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
      (span ℤ
        (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
          (⇑(fractionalIdealLatticeBasis K I :
                Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                  (mixedSpace K)) :
            (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) → mixedSpace K))))
    ((mixedEmbedding K : (a : K) → mixedSpace K) a : mixedSpace K)
h_nz :
  Ne (α :=
    ↥(toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
        (span ℤ
          (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
            (⇑(fractionalIdealLatticeBasis K I :
                  Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                    (mixedSpace K)) :
              (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) → mixedSpace K)))))
    (Subtype.mk (p := fun (x : mixedSpace K) ↦
        Membership.mem (γ := AddSubgroup (mixedSpace K))
          (toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
            (span ℤ
              (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
                (⇑(fractionalIdealLatticeBasis K I :
                      Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                        (mixedSpace K)) :
                  (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) →
                    mixedSpace K))))
          x)
        ((mixedEmbedding K : (a : K) → mixedSpace K) a) hx :
      ↥(toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
          (span ℤ
            (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
              (⇑(fractionalIdealLatticeBasis K I :
                    Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                      (mixedSpace K)) :
                (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) →
                  mixedSpace K)))))
    (0 :
      ↥(toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
          (span ℤ
            (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
              (⇑(fractionalIdealLatticeBasis K I :
                    Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                      (mixedSpace K)) :
                (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) →
                  mixedSpace K)))))
h_mem :
  Membership.mem (α := mixedSpace K) (convexBodyLT K f)
    ((↑(Subtype.mk (p := fun (x : mixedSpace K) ↦
            Membership.mem (γ := AddSubgroup (mixedSpace K))
              (toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
                (span ℤ
                  (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
                    (⇑(fractionalIdealLatticeBasis K I :
                          Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                            (mixedSpace K)) :
                      (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) →
                        mixedSpace K))))
              x)
            ((mixedEmbedding K : (a : K) → mixedSpace K) a) hx) :
        mixedSpace K) :
      mixedSpace K)
⊢ ∃ a ∈ (↑I : FractionalIdeal (𝓞 K)⁰ K), a ≠ 0 ∧ ∀ (w : InfinitePlace K), (w : (a : K) → ℝ) a < (↑(f w) : ℝ)

Tactic:
exact ⟨a, ha, by simpa using h_nz, (convexBodyLT_mem K f).mp h_mem⟩