Theorem: NumberField.mixedEmbedding.det_basisOfFractionalIdeal_eq_norm
File: Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean

Proof steps:
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
⊢ |(latticeBasis K).det (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e)| = ↑(FractionalIdeal.absNorm ↑I)
state_after: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
⊢ (latticeBasis K).det (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e) =
    (algebraMap ℚ ℝ) ((integralBasis K).det (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))
tactic: suffices Basis.det (latticeBasis K) ((mixedEmbedding K ∘ (basisOfFractionalIdeal K I) ∘ e)) =
      (algebraMap ℚ ℝ) ((Basis.det (integralBasis K)) ((basisOfFractionalIdeal K I) ∘ e)) by
    rw [this, eq_ratCast, ← Rat.cast_abs, ← Equiv.symm_symm e, ← Basis.coe_reindex,
      det_basisOfFractionalIdeal_eq_absNorm K I e]
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
⊢ (latticeBasis K).det (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e) =
    (algebraMap ℚ ℝ) ((integralBasis K).det (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))
state_after: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
⊢ ((latticeBasis K).toMatrix (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e)).det =
    ((algebraMap ℚ ℝ).mapMatrix ((integralBasis K).toMatrix (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))).det
tactic: rw [Basis.det_apply, Basis.det_apply, RingHom.map_det]
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
⊢ ((latticeBasis K).toMatrix (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e)).det =
    ((algebraMap ℚ ℝ).mapMatrix ((integralBasis K).toMatrix (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))).det
state_after: case e_M
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
⊢ (latticeBasis K).toMatrix (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e) =
    (algebraMap ℚ ℝ).mapMatrix ((integralBasis K).toMatrix (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))
tactic: congr
state_before: case e_M
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
⊢ (latticeBasis K).toMatrix (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e) =
    (algebraMap ℚ ℝ).mapMatrix ((integralBasis K).toMatrix (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))
state_after: case e_M.a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
i j : ChooseBasisIndex ℤ (𝓞 K)
⊢ (latticeBasis K).toMatrix (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e) i j =
    (algebraMap ℚ ℝ).mapMatrix ((integralBasis K).toMatrix (⇑(basisOfFractionalIdeal K I) ∘ ⇑e)) i j
tactic: ext i j
state_before: case e_M.a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
i j : ChooseBasisIndex ℤ (𝓞 K)
⊢ (latticeBasis K).toMatrix (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e) i j =
    (algebraMap ℚ ℝ).mapMatrix ((integralBasis K).toMatrix (⇑(basisOfFractionalIdeal K I) ∘ ⇑e)) i j
state_after: case e_M.a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
i j : ChooseBasisIndex ℤ (𝓞 K)
⊢ ((latticeBasis K).repr ((mixedEmbedding K) ((basisOfFractionalIdeal K I) (e j)))) i =
    (algebraMap ℚ ℝ) (((integralBasis K).repr ((basisOfFractionalIdeal K I) (e j))) i)
tactic: simp_rw [RingHom.mapMatrix_apply, Matrix.map_apply, Basis.toMatrix_apply, Function.comp_apply]
state_before: case e_M.a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
i j : ChooseBasisIndex ℤ (𝓞 K)
⊢ ((latticeBasis K).repr ((mixedEmbedding K) ((basisOfFractionalIdeal K I) (e j)))) i =
    (algebraMap ℚ ℝ) (((integralBasis K).repr ((basisOfFractionalIdeal K I) (e j))) i)
state_after: no goals
tactic: exact latticeBasis_repr_apply K _ i