R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
⊢ sInf (map C '' {J | IsMaximal J}) ≤ ⊥
R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
⊢ jacobson ⊥ = ⊥
refine' eq_bot_iff.2 (le_trans jacobson_bot_polynomial_le_sInf_map_maximal _)
R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
f : R[X]
hf : f ∈ sInf (map C '' {J | IsMaximal J})
n : ℕ
⊢ coeff f n = 0
R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
⊢ sInf (map C '' {J | IsMaximal J}) ≤ ⊥
refine' fun f hf => (Submodule.mem_bot R[X]).2 <| Polynomial.ext fun n =>
  Trans.trans (?_ : coeff f n = 0) (coeff_zero n).symm
R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
f : R[X]
hf : f ∈ sInf (map C '' {J | IsMaximal J})
n : ℕ
⊢ coeff f n ∈ jacobson ⊥
R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
f : R[X]
hf : f ∈ sInf (map C '' {J | IsMaximal J})
n : ℕ
⊢ coeff f n = 0
suffices f.coeff n ∈ Ideal.jacobson ⊥ by rwa [h, Submodule.mem_bot] at this
no goals
R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
f : R[X]
hf : f ∈ sInf (map C '' {J | IsMaximal J})
n : ℕ
⊢ coeff f n ∈ jacobson ⊥
exact mem_sInf.2 fun j hj => (mem_map_C_iff.1 ((mem_sInf.1 hf) ⟨j, ⟨hj.2, rfl⟩⟩)) n
no goals
R : Type u
S : Type v
inst✝ : CommRing R
h : jacobson ⊥ = ⊥
f : R[X]
hf : f ∈ sInf (map C '' {J | IsMaximal J})
n : ℕ
this : coeff f n ∈ jacobson ⊥
⊢ coeff f n = 0
rwa [h, Submodule.mem_bot] at this