Theorem: bind₁_rename_expand_wittPolynomial
File: Mathlib/RingTheory/WittVector/StructurePolynomial.lean

tactic: apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
state_before: p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ (bind₁ fun b => (rename fun i => (b, i)) ((expand p) (W_ ℤ n))) Φ =
    (bind₁ fun i => (expand p) (wittStructureInt p Φ i)) (W_ ℤ n)
state_after: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ (map (Int.castRingHom ℚ)) ((bind₁ fun b => (rename fun i => (b, i)) ((expand p) (W_ ℤ n))) Φ) =
    (map (Int.castRingHom ℚ)) ((bind₁ fun i => (expand p) (wittStructureInt p Φ i)) (W_ ℤ n))

tactic: simp only [map_bind₁, map_rename, map_expand, rename_expand, map_wittPolynomial]
state_before: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ (map (Int.castRingHom ℚ)) ((bind₁ fun b => (rename fun i => (b, i)) ((expand p) (W_ ℤ n))) Φ) =
    (map (Int.castRingHom ℚ)) ((bind₁ fun i => (expand p) (wittStructureInt p Φ i)) (W_ ℤ n))
state_after: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ (bind₁ fun i => (expand p) ((rename fun i_1 => (i, i_1)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)

tactic: have key := (wittStructureRat_prop p (map (Int.castRingHom ℚ) Φ) n).symm
state_before: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ (bind₁ fun i => (expand p) ((rename fun i_1 => (i, i_1)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)
state_after: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
key :
  (bind₁ fun i => (rename (Prod.mk i)) (W_ ℚ n)) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n)
⊢ (bind₁ fun i => (expand p) ((rename fun i_1 => (i, i_1)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)

tactic: apply_fun expand p at key
state_before: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
key :
  (bind₁ fun i => (rename (Prod.mk i)) (W_ ℚ n)) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n)
⊢ (bind₁ fun i => (expand p) ((rename fun i_1 => (i, i_1)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)
state_after: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
key :
  (expand p) ((bind₁ fun i => (rename (Prod.mk i)) (W_ ℚ n)) ((map (Int.castRingHom ℚ)) Φ)) =
    (expand p) ((bind₁ (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n))
⊢ (bind₁ fun i => (expand p) ((rename fun i_1 => (i, i_1)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)

tactic: simp only [expand_bind₁] at key
state_before: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
key :
  (expand p) ((bind₁ fun i => (rename (Prod.mk i)) (W_ ℚ n)) ((map (Int.castRingHom ℚ)) Φ)) =
    (expand p) ((bind₁ (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n))
⊢ (bind₁ fun i => (expand p) ((rename fun i_1 => (i, i_1)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)
state_after: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
key :
  (bind₁ fun i => (expand p) ((rename (Prod.mk i)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ fun i => (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n)
⊢ (bind₁ fun i => (expand p) ((rename fun i_1 => (i, i_1)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)

tactic: rw [key]
state_before: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
key :
  (bind₁ fun i => (expand p) ((rename (Prod.mk i)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ fun i => (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n)
⊢ (bind₁ fun i => (expand p) ((rename fun i_1 => (i, i_1)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)
state_after: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
key :
  (bind₁ fun i => (expand p) ((rename (Prod.mk i)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ fun i => (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n)
⊢ (bind₁ fun i => (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n) =
    (bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)

tactic: clear key
state_before: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
key :
  (bind₁ fun i => (expand p) ((rename (Prod.mk i)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
    (bind₁ fun i => (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n)
⊢ (bind₁ fun i => (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n) =
    (bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)
state_after: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ (bind₁ fun i => (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n) =
    (bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)

tactic: apply eval₂Hom_congr' rfl _ rfl
state_before: case a
p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ (bind₁ fun i => (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n) =
    (bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)
state_after: p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ ∀ i ∈ (W_ ℚ n).vars,
    i ∈ (W_ ℚ n).vars →
      (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i) =
        (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))

tactic: rintro i hi -
state_before: p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ ∀ i ∈ (W_ ℚ n).vars,
    i ∈ (W_ ℚ n).vars →
      (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i) =
        (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))
state_after: p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
i : ℕ
hi : i ∈ (W_ ℚ n).vars
⊢ (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i) =
    (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))

tactic: rw [wittPolynomial_vars, Finset.mem_range] at hi
state_before: p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
i : ℕ
hi : i ∈ (W_ ℚ n).vars
⊢ (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i) =
    (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))
state_after: p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
i : ℕ
hi : i < n + 1
⊢ (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i) =
    (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))

tactic: simp only [IH i hi]
state_before: p : ℕ
R : Type u_1
idx : Type u_2
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
  ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
i : ℕ
hi : i < n + 1
⊢ (expand p) (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i) =
    (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))
state_after: no goals