Module: Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic

Goal state before:
pp: case nontrivial
n : Type u_1
R : Type u_2
inst✝¹ : CommRing R
c : n →₀ R
inst✝ : IsDomain R
hc : c.support.Nontrivial
h_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r
ι : n ↪ n ⊕ n →₀ ℕ := ⋯
aux : sumSMulXSMulY c = AddMonoidAlgebra.ofCoeff (Finsupp.embDomain ι c)
hcoeff : ∀ (i : n), coeff (ι i) (sumSMulXSMulY c) = c i
hsupp : (sumSMulXSMulY c).support = Finset.map ι c.support
a : n
ha : a ∈ c.support
⊢ (sumSMulXSMulY c).support.Nontrivial

pp: case hd
n : Type u_1
R : Type u_2
inst✝¹ : CommRing R
c : n →₀ R
inst✝ : IsDomain R
hc : c.support.Nontrivial
h_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r
ι : n ↪ n ⊕ n →₀ ℕ := ⋯
aux : sumSMulXSMulY c = AddMonoidAlgebra.ofCoeff (Finsupp.embDomain ι c)
hcoeff : ∀ (i : n), coeff (ι i) (sumSMulXSMulY c) = c i
hsupp : (sumSMulXSMulY c).support = Finset.map ι c.support
a : n
ha : a ∈ c.support
⊢ ι a ∈ (sumSMulXSMulY c).support

pp: case hdi
n : Type u_1
R : Type u_2
inst✝¹ : CommRing R
c : n →₀ R
inst✝ : IsDomain R
hc : c.support.Nontrivial
h_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r
ι : n ↪ n ⊕ n →₀ ℕ := ⋯
aux : sumSMulXSMulY c = AddMonoidAlgebra.ofCoeff (Finsupp.embDomain ι c)
hcoeff : ∀ (i : n), coeff (ι i) (sumSMulXSMulY c) = c i
hsupp : (sumSMulXSMulY c).support = Finset.map ι c.support
a : n
ha : a ∈ c.support
⊢ (ι a) (Sum.inl a) = 1

pp: case disjoint
n : Type u_1
R : Type u_2
inst✝¹ : CommRing R
c : n →₀ R
inst✝ : IsDomain R
hc : c.support.Nontrivial
h_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r
ι : n ↪ n ⊕ n →₀ ℕ := ⋯
aux : sumSMulXSMulY c = AddMonoidAlgebra.ofCoeff (Finsupp.embDomain ι c)
hcoeff : ∀ (i : n), coeff (ι i) (sumSMulXSMulY c) = c i
hsupp : (sumSMulXSMulY c).support = Finset.map ι c.support
a : n
ha : a ∈ c.support
⊢ (↑(sumSMulXSMulY c).support).PairwiseDisjoint Finsupp.support

pp: case isPrimitive
n : Type u_1
R : Type u_2
inst✝¹ : CommRing R
c : n →₀ R
inst✝ : IsDomain R
hc : c.support.Nontrivial
h_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r
ι : n ↪ n ⊕ n →₀ ℕ := ⋯
aux : sumSMulXSMulY c = AddMonoidAlgebra.ofCoeff (Finsupp.embDomain ι c)
hcoeff : ∀ (i : n), coeff (ι i) (sumSMulXSMulY c) = c i
hsupp : (sumSMulXSMulY c).support = Finset.map ι c.support
a : n
ha : a ∈ c.support
⊢ ∀ (r : R), (∀ (d : n ⊕ n →₀ ℕ), r ∣ coeff d (sumSMulXSMulY c)) → IsUnit r

Tactic:
· rwa [hsupp, Finset.map_nontrivial]

Goal state after:
case hd
n : Type u_1
R : Type u_2
inst✝¹ : CommRing R
c : n →₀ R
inst✝ : IsDomain R
hc : c.support.Nontrivial
h_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r
ι : n ↪ n ⊕ n →₀ ℕ := ⋯
aux : sumSMulXSMulY c = AddMonoidAlgebra.ofCoeff (Finsupp.embDomain ι c)
hcoeff : ∀ (i : n), coeff (ι i) (sumSMulXSMulY c) = c i
hsupp : (sumSMulXSMulY c).support = Finset.map ι c.support
a : n
ha : a ∈ c.support
⊢ ι a ∈ (sumSMulXSMulY c).support

case hdi
n : Type u_1
R : Type u_2
inst✝¹ : CommRing R
c : n →₀ R
inst✝ : IsDomain R
hc : c.support.Nontrivial
h_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r
ι : n ↪ n ⊕ n →₀ ℕ := ⋯
aux : sumSMulXSMulY c = AddMonoidAlgebra.ofCoeff (Finsupp.embDomain ι c)
hcoeff : ∀ (i : n), coeff (ι i) (sumSMulXSMulY c) = c i
hsupp : (sumSMulXSMulY c).support = Finset.map ι c.support
a : n
ha : a ∈ c.support
⊢ (ι a) (Sum.inl a) = 1

case disjoint
n : Type u_1
R : Type u_2
inst✝¹ : CommRing R
c : n →₀ R
inst✝ : IsDomain R
hc : c.support.Nontrivial
h_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r
ι : n ↪ n ⊕ n →₀ ℕ := ⋯
aux : sumSMulXSMulY c = AddMonoidAlgebra.ofCoeff (Finsupp.embDomain ι c)
hcoeff : ∀ (i : n), coeff (ι i) (sumSMulXSMulY c) = c i
hsupp : (sumSMulXSMulY c).support = Finset.map ι c.support
a : n
ha : a ∈ c.support
⊢ (↑(sumSMulXSMulY c).support).PairwiseDisjoint Finsupp.support

case isPrimitive
n : Type u_1
R : Type u_2
inst✝¹ : CommRing R
c : n →₀ R
inst✝ : IsDomain R
hc : c.support.Nontrivial
h_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r
ι : n ↪ n ⊕ n →₀ ℕ := ⋯
aux : sumSMulXSMulY c = AddMonoidAlgebra.ofCoeff (Finsupp.embDomain ι c)
hcoeff : ∀ (i : n), coeff (ι i) (sumSMulXSMulY c) = c i
hsupp : (sumSMulXSMulY c).support = Finset.map ι c.support
a : n
ha : a ∈ c.support
⊢ ∀ (r : R), (∀ (d : n ⊕ n →₀ ℕ), r ∣ coeff d (sumSMulXSMulY c)) → IsUnit r