-- 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
w₀ : { w : InfinitePlace K // w.IsComplex }
h : minkowskiBound K I < ((volume : Measure (mixedSpace K)) : (a : Set (mixedSpace K)) → ℝ≥0∞) (convexBodyLT' K f w₀)
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 :
      (↑(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) ∈
        convexBodyLT' K f w₀),
    ∃ a ∈ (↑I : FractionalIdeal (𝓞 K)⁰ K),
      a ≠ 0 ∧
        (∀ (w : InfinitePlace K), w ≠ (↑w₀ : InfinitePlace K) → (w : (a : K) → ℝ) a < (↑(f w) : ℝ)) ∧
          |Complex.re (((↑w₀ : InfinitePlace K).embedding : (a : K) → ℂ) a)| < 1 ∧
            |Complex.im (((↑w₀ : InfinitePlace K).embedding : (a : K) → ℂ) a)| <
              HPow.hPow (β := ℕ) (↑(f (↑w₀ : InfinitePlace K)) : ℝ) 2
⊢ ∃ a ∈ (↑I : FractionalIdeal (𝓞 K)⁰ K),
    a ≠ 0 ∧
      (∀ (w : InfinitePlace K), w ≠ (↑w₀ : InfinitePlace K) → (w : (a : K) → ℝ) a < (↑(f w) : ℝ)) ∧
        |Complex.re (((↑w₀ : InfinitePlace K).embedding : (a : K) → ℂ) a)| < 1 ∧
          |Complex.im (((↑w₀ : InfinitePlace K).embedding : (a : K) → ℂ) a)| <
            HPow.hPow (β := ℕ) (↑(f (↑w₀ : InfinitePlace K)) : ℝ) 2

Tactic:
obtain ⟨⟨x, hx⟩, h_nz, h_mem⟩ := exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure
    h_fund (convexBodyLT'_neg_mem K f w₀) (convexBodyLT'_convex K f w₀) h

Premises used: MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure, NumberField.mixedEmbedding.convexBodyLT'_neg_mem, NumberField.mixedEmbedding.convexBodyLT'_convex