Theorem: char_dvd_card_solutions_of_sum_lt
File: Mathlib/FieldTheory/ChevalleyWarning.lean

Proof steps:
state_before: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
⊢ p ∣ Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 }
state_after: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
⊢ p ∣ Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 }
tactic: have hq : 0 < q - 1 := by rw [← Fintype.card_units, Fintype.card_pos_iff]; exact ⟨1⟩
state_before: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
⊢ p ∣ Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 }
state_after: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
⊢ p ∣ Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 }
tactic: let S : Finset (σ → K) := { x ∈ univ | ∀ i ∈ s, eval x (f i) = 0 }.toFinset
state_before: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
⊢ p ∣ Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 }
state_after: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
⊢ p ∣ Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 }
tactic: have hS : ∀ x : σ → K, x ∈ S ↔ ∀ i : ι, i ∈ s → eval x (f i) = 0 := by
    intro x
    simp only [S, Set.toFinset_setOf, mem_univ, true_and, mem_filter]
state_before: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
⊢ p ∣ Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 }
state_after: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
F : MvPolynomial σ K := ∏ i ∈ s, (1 - f i ^ (q - 1))
⊢ p ∣ Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 }
tactic: let F : MvPolynomial σ K := ∏ i ∈ s, (1 - f i ^ (q - 1))
state_before: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
F : MvPolynomial σ K := ∏ i ∈ s, (1 - f i ^ (q - 1))
hF : ∀ (x : σ → K), (eval x) F = if x ∈ S then 1 else 0
⊢ p ∣ Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 }
state_after: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
F : MvPolynomial σ K := ∏ i ∈ s, (1 - f i ^ (q - 1))
hF : ∀ (x : σ → K), (eval x) F = if x ∈ S then 1 else 0
key : ∑ x : σ → K, (eval x) F = ↑(Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 })
⊢ p ∣ Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 }
tactic: have key : ∑ x, eval x F = Fintype.card { x : σ → K // ∀ i ∈ s, eval x (f i) = 0 } := by
    rw [Fintype.card_of_subtype S hS, card_eq_sum_ones, Nat.cast_sum, Nat.cast_one, ←
      Fintype.sum_extend_by_zero S, sum_congr rfl fun x _ => hF x]
state_before: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
F : MvPolynomial σ K := ∏ i ∈ s, (1 - f i ^ (q - 1))
hF : ∀ (x : σ → K), (eval x) F = if x ∈ S then 1 else 0
key : ∑ x : σ → K, (eval x) F = ↑(Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 })
⊢ p ∣ Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 }
state_after: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
F : MvPolynomial σ K := ∏ i ∈ s, (1 - f i ^ (q - 1))
hF : ∀ (x : σ → K), (eval x) F = if x ∈ S then 1 else 0
key : ∑ x : σ → K, (eval x) F = ↑(Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 })
⊢ ∑ x : σ → K, (eval x) F = 0
tactic: rw [← CharP.cast_eq_zero_iff K, ← key]
state_before: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
F : MvPolynomial σ K := ∏ i ∈ s, (1 - f i ^ (q - 1))
hF : ∀ (x : σ → K), (eval x) F = if x ∈ S then 1 else 0
key : ∑ x : σ → K, (eval x) F = ↑(Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 })
⊢ ∑ x : σ → K, (eval x) F = 0
state_after: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
F : MvPolynomial σ K := ∏ i ∈ s, (1 - f i ^ (q - 1))
hF : ∀ (x : σ → K), (eval x) F = if x ∈ S then 1 else 0
key : ∑ x : σ → K, (eval x) F = ↑(Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 })
⊢ F.totalDegree < (q - 1) * Fintype.card σ
tactic: apply F.sum_eval_eq_zero
state_before: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
F : MvPolynomial σ K := ∏ i ∈ s, (1 - f i ^ (q - 1))
hF : ∀ (x : σ → K), (eval x) F = if x ∈ S then 1 else 0
key : ∑ x : σ → K, (eval x) F = ↑(Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 })
⊢ F.totalDegree < (q - 1) * Fintype.card σ
state_after: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
F : MvPolynomial σ K := ∏ i ∈ s, (1 - f i ^ (q - 1))
hF : ∀ (x : σ → K), (eval x) F = if x ∈ S then 1 else 0
key : ∑ x : σ → K, (eval x) F = ↑(Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 })
i : ι
x✝ : i ∈ s
⊢ (1 - f i ^ (q - 1)).totalDegree ≤ (q - 1) * (f i).totalDegree
tactic: calc
    F.totalDegree ≤ ∑ i ∈ s, (1 - f i ^ (q - 1)).totalDegree := totalDegree_finset_prod s _
    _ ≤ ∑ i ∈ s, (q - 1) * (f i).totalDegree := sum_le_sum fun i _ => ?_
    -- see ↓
    _ = (q - 1) * ∑ i ∈ s, (f i).totalDegree := (mul_sum ..).symm
    _ < (q - 1) * Fintype.card σ := by rwa [mul_lt_mul_left hq]
state_before: K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
F : MvPolynomial σ K := ∏ i ∈ s, (1 - f i ^ (q - 1))
hF : ∀ (x : σ → K), (eval x) F = if x ∈ S then 1 else 0
key : ∑ x : σ → K, (eval x) F = ↑(Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 })
i : ι
x✝ : i ∈ s
⊢ (1 - f i ^ (q - 1)).totalDegree ≤ (q - 1) * (f i).totalDegree
state_after: no goals
tactic: calc
    (1 - f i ^ (q - 1)).totalDegree ≤
        max (1 : MvPolynomial σ K).totalDegree (f i ^ (q - 1)).totalDegree := totalDegree_sub _ _
    _ ≤ (f i ^ (q - 1)).totalDegree := by simp
    _ ≤ (q - 1) * (f i).totalDegree := totalDegree_pow _ _