Theorem: quadraticChar_card_card
File: Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean

tactic: let χ := (quadraticChar F).ringHomComp (algebraMap ℤ F')
state_before: F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
⊢ (quadraticChar F) ↑(Fintype.card F') = (quadraticChar F') (↑((quadraticChar F) (-1)) * ↑(Fintype.card F))
state_after: F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
⊢ (quadraticChar F) ↑(Fintype.card F') = (quadraticChar F') (↑((quadraticChar F) (-1)) * ↑(Fintype.card F))

tactic: have hχ₁ : χ ≠ 1 := by
  obtain ⟨a, ha⟩ := quadraticChar_exists_neg_one' hF
  refine ne_one_iff.mpr ⟨a, ?_⟩
  simpa only [ringHomComp_apply, ha, eq_intCast, Int.cast_neg, Int.cast_one, χ] using
    Ring.neg_one_ne_one_of_char_ne_two hF'
state_before: F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
⊢ (quadraticChar F) ↑(Fintype.card F') = (quadraticChar F') (↑((quadraticChar F) (-1)) * ↑(Fintype.card F))
state_after: F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
hχ₁ : χ ≠ 1
⊢ (quadraticChar F) ↑(Fintype.card F') = (quadraticChar F') (↑((quadraticChar F) (-1)) * ↑(Fintype.card F))

tactic: have h := Char.card_pow_card hχ₁ ((quadraticChar_isQuadratic F).comp _) h hF'
state_before: F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
hχ₁ : χ ≠ 1
⊢ (quadraticChar F) ↑(Fintype.card F') = (quadraticChar F') (↑((quadraticChar F) (-1)) * ↑(Fintype.card F))
state_after: F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h✝ : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
hχ₁ : χ ≠ 1
h : (χ (-1) * ↑(Fintype.card F)) ^ (Fintype.card F' / 2) = χ ↑(Fintype.card F')
⊢ (quadraticChar F) ↑(Fintype.card F') = (quadraticChar F') (↑((quadraticChar F) (-1)) * ↑(Fintype.card F))

tactic: rw [← quadraticChar_eq_pow_of_char_ne_two' hF'] at h
state_before: F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h✝ : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
hχ₁ : χ ≠ 1
h : (χ (-1) * ↑(Fintype.card F)) ^ (Fintype.card F' / 2) = χ ↑(Fintype.card F')
⊢ (quadraticChar F) ↑(Fintype.card F') = (quadraticChar F') (↑((quadraticChar F) (-1)) * ↑(Fintype.card F))
state_after: F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h✝ : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
hχ₁ : χ ≠ 1
h : ↑((quadraticChar F') (χ (-1) * ↑(Fintype.card F))) = χ ↑(Fintype.card F')
⊢ (quadraticChar F) ↑(Fintype.card F') = (quadraticChar F') (↑((quadraticChar F) (-1)) * ↑(Fintype.card F))

tactic: exact (IsQuadratic.eq_of_eq_coe (quadraticChar_isQuadratic F')
  (quadraticChar_isQuadratic F) hF' h).symm
state_before: F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h✝ : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
hχ₁ : χ ≠ 1
h : ↑((quadraticChar F') (χ (-1) * ↑(Fintype.card F))) = χ ↑(Fintype.card F')
⊢ (quadraticChar F) ↑(Fintype.card F') = (quadraticChar F') (↑((quadraticChar F) (-1)) * ↑(Fintype.card F))
state_after: no goals

tactic: obtain ⟨a, ha⟩ := quadraticChar_exists_neg_one' hF
state_before: F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
⊢ χ ≠ 1
state_after: case intro
F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
a : Fˣ
ha : (quadraticChar F) ↑a = -1
⊢ χ ≠ 1

tactic: refine ne_one_iff.mpr ⟨a, ?_⟩
state_before: case intro
F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
a : Fˣ
ha : (quadraticChar F) ↑a = -1
⊢ χ ≠ 1
state_after: case intro
F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
a : Fˣ
ha : (quadraticChar F) ↑a = -1
⊢ χ ↑a ≠ 1

tactic: simpa only [ringHomComp_apply, ha, eq_intCast, Int.cast_neg, Int.cast_one, χ] using
  Ring.neg_one_ne_one_of_char_ne_two hF'
state_before: case intro
F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
a : Fˣ
ha : (quadraticChar F) ↑a = -1
⊢ χ ↑a ≠ 1
state_after: no goals