Theorem: WittVector.poly_eq_of_wittPolynomial_bind_eq
File: Mathlib/RingTheory/WittVector/IsPoly.lean

tactic: ext1 n
state_before: p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial ℕ ℤ
h : ∀ (n : ℕ), (bind₁ f) (wittPolynomial p ℤ n) = (bind₁ g) (wittPolynomial p ℤ n)
⊢ f = g
state_after: case h
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial ℕ ℤ
h : ∀ (n : ℕ), (bind₁ f) (wittPolynomial p ℤ n) = (bind₁ g) (wittPolynomial p ℤ n)
n : ℕ
⊢ f n = g n

tactic: apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
state_before: case h
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial ℕ ℤ
h : ∀ (n : ℕ), (bind₁ f) (wittPolynomial p ℤ n) = (bind₁ g) (wittPolynomial p ℤ n)
n : ℕ
⊢ f n = g n
state_after: case h.a
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial ℕ ℤ
h : ∀ (n : ℕ), (bind₁ f) (wittPolynomial p ℤ n) = (bind₁ g) (wittPolynomial p ℤ n)
n : ℕ
⊢ (MvPolynomial.map (Int.castRingHom ℚ)) (f n) = (MvPolynomial.map (Int.castRingHom ℚ)) (g n)

tactic: rw [← Function.funext_iff] at h
state_before: case h.a
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial ℕ ℤ
h : ∀ (n : ℕ), (bind₁ f) (wittPolynomial p ℤ n) = (bind₁ g) (wittPolynomial p ℤ n)
n : ℕ
⊢ (MvPolynomial.map (Int.castRingHom ℚ)) (f n) = (MvPolynomial.map (Int.castRingHom ℚ)) (g n)
state_after: case h.a
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial ℕ ℤ
h : (fun n => (bind₁ f) (wittPolynomial p ℤ n)) = fun n => (bind₁ g) (wittPolynomial p ℤ n)
n : ℕ
⊢ (MvPolynomial.map (Int.castRingHom ℚ)) (f n) = (MvPolynomial.map (Int.castRingHom ℚ)) (g n)

tactic: replace h :=
  congr_arg (fun fam => bind₁ (MvPolynomial.map (Int.castRingHom ℚ) ∘ fam) (xInTermsOfW p ℚ n)) h
state_before: case h.a
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial ℕ ℤ
h : (fun n => (bind₁ f) (wittPolynomial p ℤ n)) = fun n => (bind₁ g) (wittPolynomial p ℤ n)
n : ℕ
⊢ (MvPolynomial.map (Int.castRingHom ℚ)) (f n) = (MvPolynomial.map (Int.castRingHom ℚ)) (g n)
state_after: case h.a
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial ℕ ℤ
n : ℕ
h :
  ((fun fam => (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fam)) (xInTermsOfW p ℚ n)) fun n =>
      (bind₁ f) (wittPolynomial p ℤ n)) =
    (fun fam => (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fam)) (xInTermsOfW p ℚ n)) fun n =>
      (bind₁ g) (wittPolynomial p ℤ n)
⊢ (MvPolynomial.map (Int.castRingHom ℚ)) (f n) = (MvPolynomial.map (Int.castRingHom ℚ)) (g n)

tactic: simpa only [Function.comp, map_bind₁, map_wittPolynomial, ← bind₁_bind₁,
  bind₁_wittPolynomial_xInTermsOfW, bind₁_X_right] using h
state_before: case h.a
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial ℕ ℤ
n : ℕ
h :
  ((fun fam => (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fam)) (xInTermsOfW p ℚ n)) fun n =>
      (bind₁ f) (wittPolynomial p ℤ n)) =
    (fun fam => (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fam)) (xInTermsOfW p ℚ n)) fun n =>
      (bind₁ g) (wittPolynomial p ℤ n)
⊢ (MvPolynomial.map (Int.castRingHom ℚ)) (f n) = (MvPolynomial.map (Int.castRingHom ℚ)) (g n)
state_after: no goals