-- Mathlib/NumberTheory/NumberField/Discriminant/Defs.lean
theorem Algebra.discr_eq_discr_of_toMatrix_coeff_isIntegral [NumberField K]
    {b : Basis ι ℚ K} {b' : Basis ι' ℚ K} (h : ∀ i j, IsIntegral ℤ (b.toMatrix b' i j))
    (h' : ∀ i j, IsIntegral ℤ (b'.toMatrix b i j)) : discr ℚ b = discr ℚ b'

Proof state:
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i : ι') (j : ι), IsIntegral ℤ (b'.toMatrix (⇑b) i j)
i j : ι'
⊢ IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)

Tactic:
convert h' i ((b.indexEquiv b').symm j)

Result:
case h.e'_6
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i : ι') (j : ι), IsIntegral ℤ (b'.toMatrix (⇑b) i j)
i j : ι'
⊢ b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j = b'.toMatrix (⇑b) i ((b.indexEquiv b').symm j)

Proof state:
case intro
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
r : ℤ
hr : (algebraMap ℤ ℚ) r = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
this : IsIntegral ℤ (b'.toMatrix ⇑(b.reindex (b.indexEquiv b'))).det
r' : ℤ
hr' : (algebraMap ℤ ℚ) r' = (b'.toMatrix ⇑(b.reindex (b.indexEquiv b'))).det
⊢ (algebraMap ℤ ℚ) (r * r') = 1

Tactic:
rw [RingHom.map_mul, hr, hr', ← Matrix.det_mul,
  Basis.toMatrix_mul_toMatrix_flip, Matrix.det_one]

Result:
no goals

Proof state:
case intro
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
r : ℤ
hr : (algebraMap ℤ ℚ) r = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
this : IsIntegral ℤ (b'.toMatrix ⇑(b.reindex (b.indexEquiv b'))).det
r' : ℤ
hr' : (algebraMap ℤ ℚ) r' = (b'.toMatrix ⇑(b.reindex (b.indexEquiv b'))).det
⊢ IsUnit r

Tactic:
refine isUnit_iff_exists_inv.2 ⟨r', ?_⟩

Result:
case intro
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
r : ℤ
hr : (algebraMap ℤ ℚ) r = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
this : IsIntegral ℤ (b'.toMatrix ⇑(b.reindex (b.indexEquiv b'))).det
r' : ℤ
hr' : (algebraMap ℤ ℚ) r' = (b'.toMatrix ⇑(b.reindex (b.indexEquiv b'))).det
⊢ r * r' = 1

Proof state:
case intro
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
r : ℤ
hr : (algebraMap ℤ ℚ) r = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
this : IsIntegral ℤ (b'.toMatrix ⇑(b.reindex (b.indexEquiv b'))).det
r' : ℤ
hr' : (algebraMap ℤ ℚ) r' = (b'.toMatrix ⇑(b.reindex (b.indexEquiv b'))).det
⊢ r * r' = 1

Tactic:
suffices algebraMap ℤ ℚ (r * r') = 1 by
  rw [← RingHom.map_one (algebraMap ℤ ℚ)] at this
  exact (IsFractionRing.injective ℤ ℚ) this

Result:
case intro
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
r : ℤ
hr : (algebraMap ℤ ℚ) r = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
this : IsIntegral ℤ (b'.toMatrix ⇑(b.reindex (b.indexEquiv b'))).det
r' : ℤ
hr' : (algebraMap ℤ ℚ) r' = (b'.toMatrix ⇑(b.reindex (b.indexEquiv b'))).det
⊢ (algebraMap ℤ ℚ) (r * r') = 1

Proof state:
case e_a
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
⊢ 1 = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det ^ 2

Tactic:
have hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix b').det :=
  IsIntegral.det fun i j => h _ _

Result:
case e_a
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
⊢ 1 = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det ^ 2

Proof state:
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
r : ℤ
hr : (algebraMap ℤ ℚ) r = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
this✝ : IsIntegral ℤ (b'.toMatrix ⇑(b.reindex (b.indexEquiv b'))).det
r' : ℤ
hr' : (algebraMap ℤ ℚ) r' = (b'.toMatrix ⇑(b.reindex (b.indexEquiv b'))).det
this : (algebraMap ℤ ℚ) (r * r') = (algebraMap ℤ ℚ) 1
⊢ r * r' = 1

Tactic:
exact (IsFractionRing.injective ℤ ℚ) this

Result:
no goals