-- Mathlib/Algebra/Module/ZLattice/Basic.lean
theorem measure_fundamentalDomain [Fintype ι] [DecidableEq ι] [MeasurableSpace E] (μ : Measure E)
    [BorelSpace E] [Measure.IsAddHaarMeasure μ] (b₀ : Basis ι ℝ E) :
    μ (fundamentalDomain b) = ENNReal.ofReal |b₀.det b| * μ (fundamentalDomain b₀)

Proof state:
case h.e'_3.h.e'_5.h.e'_1.h.e'_4.e_M
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
⊢ b₀.toMatrix ⇑b = (LinearMap.toMatrix b₀ b₀) ↑(b₀.equiv b (Equiv.refl ι))

Tactic:
ext

Result:
case h.e'_3.h.e'_5.h.e'_1.h.e'_4.e_M.a
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
i✝ j✝ : ι
⊢ b₀.toMatrix (⇑b) i✝ j✝ = (LinearMap.toMatrix b₀ b₀) (↑(b₀.equiv b (Equiv.refl ι))) i✝ j✝

Proof state:
case h.e'_3.h.e'_5.h.e'_1.h.e'_4
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
⊢ b₀.det ⇑b = LinearMap.det ↑(b.equiv b₀ (Equiv.refl ι)).symm

Tactic:
rw [← LinearMap.det_toMatrix b₀, Basis.equiv_symm, Equiv.refl_symm, Basis.det_apply]

Result:
case h.e'_3.h.e'_5.h.e'_1.h.e'_4
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
⊢ (b₀.toMatrix ⇑b).det = ((LinearMap.toMatrix b₀ b₀) ↑(b₀.equiv b (Equiv.refl ι))).det

Proof state:
case h.e'_3.h.e'_5.h.e'_1.h.e'_4.e_M.a
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
i✝ j✝ : ι
⊢ b₀.toMatrix (⇑b) i✝ j✝ = (LinearMap.toMatrix b₀ b₀) (↑(b₀.equiv b (Equiv.refl ι))) i✝ j✝

Tactic:
simp [Basis.toMatrix_apply, LinearMap.toMatrix_apply, LinearEquiv.coe_coe, Basis.equiv_apply]

Result:
no goals

Proof state:
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
⊢ μ (fundamentalDomain b) = ENNReal.ofReal |b₀.det ⇑b| * μ (fundamentalDomain b₀)

Tactic:
convert μ.addHaar_preimage_linearEquiv (b.equiv b₀ (Equiv.refl ι)) (fundamentalDomain b₀)

Result:
case h.e'_2.h.e'_6
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
⊢ fundamentalDomain b = ⇑(b.equiv b₀ (Equiv.refl ι)) ⁻¹' fundamentalDomain b₀

case h.e'_3.h.e'_5.h.e'_1.h.e'_4
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
⊢ b₀.det ⇑b = LinearMap.det ↑(b.equiv b₀ (Equiv.refl ι)).symm

Proof state:
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
⊢ μ (fundamentalDomain b) = ENNReal.ofReal |b₀.det ⇑b| * μ (fundamentalDomain b₀)

Tactic:
have : FiniteDimensional ℝ E := FiniteDimensional.of_fintype_basis b

Result:
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
⊢ μ (fundamentalDomain b) = ENNReal.ofReal |b₀.det ⇑b| * μ (fundamentalDomain b₀)

Proof state:
case h.e'_2.h.e'_6
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
⊢ fundamentalDomain b = ⇑(b.equiv b₀ (Equiv.refl ι)) ⁻¹' fundamentalDomain b₀

Tactic:
rw [Set.eq_preimage_iff_image_eq (LinearEquiv.bijective _), map_fundamentalDomain,
  Basis.map_equiv, Equiv.refl_symm, Basis.reindex_refl]

Result:
no goals

Proof state:
case h.e'_3.h.e'_5.h.e'_1.h.e'_4
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
⊢ (b₀.toMatrix ⇑b).det = ((LinearMap.toMatrix b₀ b₀) ↑(b₀.equiv b (Equiv.refl ι))).det

Tactic:
congr

Result:
case h.e'_3.h.e'_5.h.e'_1.h.e'_4.e_M
E : Type u_1
ι : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
b₀ : Basis ι ℝ E
this : FiniteDimensional ℝ E
⊢ b₀.toMatrix ⇑b = (LinearMap.toMatrix b₀ b₀) ↑(b₀.equiv b (Equiv.refl ι))