Theorem: NumberField.mixedEmbedding.exists_ne_zero_mem_ideal_lt
File: Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean

Proof steps:
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < volume (convexBodyLT K f)
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)
state_after: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < volume (convexBodyLT K f)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)
tactic: have h_fund := Zspan.isAddFundamentalDomain (fractionalIdealLatticeBasis K I) volume
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < volume (convexBodyLT K f)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)
state_after: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < volume (convexBodyLT K f)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)
tactic: have : Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I))).toAddSubgroup := by
    change Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I)) : Set (E K))
    infer_instance
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < volume (convexBodyLT K f)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)
state_after: case intro.mk.intro
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < volume (convexBodyLT K f)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)
hx : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨x, hx⟩ ≠ 0
h_mem : ↑⟨x, hx⟩ ∈ convexBodyLT K f
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)
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) (convexBodyLT_convex K f) h
state_before: case intro.mk.intro
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < volume (convexBodyLT K f)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)
hx : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨x, hx⟩ ≠ 0
h_mem : ↑⟨x, hx⟩ ∈ convexBodyLT K f
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)
state_after: case intro.mk.intro
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < volume (convexBodyLT K f)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)
hx✝ : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
hx : x ∈ ⇑(mixedEmbedding K) '' ↑↑I
h_nz : ⟨x, hx✝⟩ ≠ 0
h_mem : ↑⟨x, hx✝⟩ ∈ convexBodyLT K f
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)
tactic: rw [mem_toAddSubgroup, mem_span_fractionalIdealLatticeBasis] at hx
state_before: case intro.mk.intro
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < volume (convexBodyLT K f)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)
hx✝ : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
hx : x ∈ ⇑(mixedEmbedding K) '' ↑↑I
h_nz : ⟨x, hx✝⟩ ≠ 0
h_mem : ↑⟨x, hx✝⟩ ∈ convexBodyLT K f
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)
state_after: case intro.mk.intro.intro.intro
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < volume (convexBodyLT K f)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
a : K
ha : a ∈ ↑↑I
hx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0
h_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodyLT K f
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)
tactic: obtain ⟨a, ha, rfl⟩ := hx
state_before: case intro.mk.intro.intro.intro
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
h : minkowskiBound K I < volume (convexBodyLT K f)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
a : K
ha : a ∈ ↑↑I
hx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0
h_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodyLT K f
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)
state_after: no goals
tactic: exact ⟨a, ha, by simpa using h_nz, (convexBodyLT_mem K f).mp h_mem⟩