Theorem: Polynomial.isUnit_primPart_C
File: Mathlib/RingTheory/Polynomial/Content.lean

tactic: by_cases h0 : r = 0
state_before: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
⊢ IsUnit (C r).primPart
state_after: case pos
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : r = 0
⊢ IsUnit (C r).primPart

case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬r = 0
⊢ IsUnit (C r).primPart

tactic: unfold IsUnit
state_before: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬r = 0
⊢ IsUnit (C r).primPart
state_after: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬r = 0
⊢ ∃ u, ↑u = (C r).primPart

tactic: refine
  ⟨⟨C ↑(normUnit r)⁻¹, C ↑(normUnit r), by rw [← RingHom.map_mul, Units.inv_mul, C_1], by
      rw [← RingHom.map_mul, Units.mul_inv, C_1]⟩,
    ?_⟩
state_before: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬r = 0
⊢ ∃ u, ↑u = (C r).primPart
state_after: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬r = 0
⊢ ↑{ val := C ↑(normUnit r)⁻¹, inv := C ↑(normUnit r), val_inv := ⋯, inv_val := ⋯ } = (C r).primPart

tactic: rw [← normalize_eq_zero, ← C_eq_zero] at h0
state_before: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬r = 0
⊢ ↑{ val := C ↑(normUnit r)⁻¹, inv := C ↑(normUnit r), val_inv := ⋯, inv_val := ⋯ } = (C r).primPart
state_after: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬C (normalize r) = 0
⊢ ↑{ val := C ↑(normUnit r)⁻¹, inv := C ↑(normUnit r), val_inv := ⋯, inv_val := ⋯ } = (C r).primPart

tactic: apply mul_left_cancel₀ h0
state_before: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬C (normalize r) = 0
⊢ ↑{ val := C ↑(normUnit r)⁻¹, inv := C ↑(normUnit r), val_inv := ⋯, inv_val := ⋯ } = (C r).primPart
state_after: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬C (normalize r) = 0
⊢ C (normalize r) * ↑{ val := C ↑(normUnit r)⁻¹, inv := C ↑(normUnit r), val_inv := ⋯, inv_val := ⋯ } =
    C (normalize r) * (C r).primPart

tactic: conv_rhs => rw [← content_C, ← (C r).eq_C_content_mul_primPart]
state_before: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬C (normalize r) = 0
⊢ C (normalize r) * ↑{ val := C ↑(normUnit r)⁻¹, inv := C ↑(normUnit r), val_inv := ⋯, inv_val := ⋯ } =
    C (normalize r) * (C r).primPart
state_after: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬C (normalize r) = 0
⊢ C (normalize r) * ↑{ val := C ↑(normUnit r)⁻¹, inv := C ↑(normUnit r), val_inv := ⋯, inv_val := ⋯ } = C r

tactic: simp only [Units.val_mk, normalize_apply, RingHom.map_mul]
state_before: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬C (normalize r) = 0
⊢ C (normalize r) * ↑{ val := C ↑(normUnit r)⁻¹, inv := C ↑(normUnit r), val_inv := ⋯, inv_val := ⋯ } = C r
state_after: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬C (normalize r) = 0
⊢ C r * C ↑(normUnit r) * C ↑(normUnit r)⁻¹ = C r

tactic: rw [mul_assoc, ← RingHom.map_mul, Units.mul_inv, C_1, mul_one]
state_before: case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬C (normalize r) = 0
⊢ C r * C ↑(normUnit r) * C ↑(normUnit r)⁻¹ = C r
state_after: no goals

tactic: simp [h0]
state_before: case pos
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : r = 0
⊢ IsUnit (C r).primPart
state_after: no goals

tactic: rw [← RingHom.map_mul, Units.inv_mul, C_1]
state_before: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬r = 0
⊢ C ↑(normUnit r)⁻¹ * C ↑(normUnit r) = 1
state_after: no goals

tactic: rw [← RingHom.map_mul, Units.mul_inv, C_1]
state_before: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
r : R
h0 : ¬r = 0
⊢ C ↑(normUnit r) * C ↑(normUnit r)⁻¹ = 1
state_after: no goals