-- Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
I : (FractionalIdeal (R := 𝓞 K) (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ ((volume : Measure (mixedSpace K)) : (a : Set (mixedSpace K)) → ℝ≥0∞) (convexBodySum K B)
hB : 0 ≤ B
h1 : LT.lt (α := ℝ) 0 (↑(finrank ℚ K) : ℝ)⁻¹
h2 : 0 ≤ B / (↑(finrank ℚ K) : ℝ)
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))))
obtain :
  ∀ (x : mixedSpace K)
    (hx :
      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)
    (h_nz :
      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)
          x hx ≠
        0)
    (h_mem :
      Membership.mem (γ := Set (mixedSpace K)) (convexBodySum K B)
        (↑(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)
              x hx) :
          mixedSpace K)),
    ∃ a ∈ (↑I : FractionalIdeal (𝓞 K)⁰ K),
      a ≠ 0 ∧ (↑|(Algebra.norm ℚ : (a : K) → ℚ) a| : ℝ) ≤ (B / (↑(finrank ℚ K) : ℝ)) ^ finrank ℚ K
⊢ ∃ a ∈ (↑I : FractionalIdeal (𝓞 K)⁰ K),
    a ≠ 0 ∧ (↑|(Algebra.norm ℚ : (a : K) → ℚ) a| : ℝ) ≤ (B / (↑(finrank ℚ K) : ℝ)) ^ finrank ℚ K

Tactic:
obtain ⟨⟨x, hx⟩, h_nz, h_mem⟩ := exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure
      h_fund (fun _ ↦ convexBodySum_neg_mem K B) (convexBodySum_convex K B)
      (convexBodySum_compact K B) h

Premises used: NumberField.mixedEmbedding.convexBodySum_neg_mem, NumberField.mixedEmbedding.convexBodySum_compact, MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure, NumberField.mixedEmbedding.convexBodySum_convex