Module: Mathlib.RingTheory.GradedAlgebra.Radical

Goal state before:
pp: ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : LinearOrder ι
inst✝³ : IsOrderedCancelAddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I
x y : A
rid₁ : x ∉ I
rid₂ : y ∉ I
set₁ : Finset ι := {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}
set₁_eq : set₁ = {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}
set₂ : Finset ι := {i ∈ DFinsupp.support ((decompose 𝒜) y) | (proj 𝒜 i) y ∉ I}
set₂_eq : set₂ = {i ∈ DFinsupp.support ((decompose 𝒜) y) | (proj 𝒜 i) y ∉ I}
nonempty : ∀ x ∉ I, {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}.Nonempty
max₁ : ι := set₁.max' ⋯
max₂ : ι := set₂.max' ⋯
mem_max₁ : max₁ ∈ set₁
mem_max₂ : max₂ ∈ set₂
hxy : (proj 𝒜 (max₁ + max₂)) (x * y) ∈ I
mem_I : (proj 𝒜 max₁) x * (proj 𝒜 max₂) y ∈ I
neither_mem : (proj 𝒜 max₁) x ∉ I ∧ (proj 𝒜 max₂) y ∉ I
_rid : (proj 𝒜 max₁) x * (proj 𝒜 max₂) y ∈ I
⊢ False

Tactic:
rcases homogeneous_mem_or_mem ⟨max₁, SetLike.coe_mem _⟩ ⟨max₂, SetLike.coe_mem _⟩ mem_I
          with h | h

Goal state after:
case inl
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : LinearOrder ι
inst✝³ : IsOrderedCancelAddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I
x y : A
rid₁ : x ∉ I
rid₂ : y ∉ I
set₁ : Finset ι := {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}
set₁_eq : set₁ = {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}
set₂ : Finset ι := {i ∈ DFinsupp.support ((decompose 𝒜) y) | (proj 𝒜 i) y ∉ I}
set₂_eq : set₂ = {i ∈ DFinsupp.support ((decompose 𝒜) y) | (proj 𝒜 i) y ∉ I}
nonempty : ∀ x ∉ I, {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}.Nonempty
max₁ : ι := set₁.max' ⋯
max₂ : ι := set₂.max' ⋯
mem_max₁ : max₁ ∈ set₁
mem_max₂ : max₂ ∈ set₂
hxy : (proj 𝒜 (max₁ + max₂)) (x * y) ∈ I
mem_I : (proj 𝒜 max₁) x * (proj 𝒜 max₂) y ∈ I
neither_mem : (proj 𝒜 max₁) x ∉ I ∧ (proj 𝒜 max₂) y ∉ I
_rid : (proj 𝒜 max₁) x * (proj 𝒜 max₂) y ∈ I
h : ↑(((DFinsupp.evalAddMonoidHom max₁).comp (decomposeRingEquiv 𝒜).toRingHom.toAddMonoidHom) x) ∈ I
⊢ False

case inr
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : LinearOrder ι
inst✝³ : IsOrderedCancelAddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I
x y : A
rid₁ : x ∉ I
rid₂ : y ∉ I
set₁ : Finset ι := {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}
set₁_eq : set₁ = {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}
set₂ : Finset ι := {i ∈ DFinsupp.support ((decompose 𝒜) y) | (proj 𝒜 i) y ∉ I}
set₂_eq : set₂ = {i ∈ DFinsupp.support ((decompose 𝒜) y) | (proj 𝒜 i) y ∉ I}
nonempty : ∀ x ∉ I, {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}.Nonempty
max₁ : ι := set₁.max' ⋯
max₂ : ι := set₂.max' ⋯
mem_max₁ : max₁ ∈ set₁
mem_max₂ : max₂ ∈ set₂
hxy : (proj 𝒜 (max₁ + max₂)) (x * y) ∈ I
mem_I : (proj 𝒜 max₁) x * (proj 𝒜 max₂) y ∈ I
neither_mem : (proj 𝒜 max₁) x ∉ I ∧ (proj 𝒜 max₂) y ∉ I
_rid : (proj 𝒜 max₁) x * (proj 𝒜 max₂) y ∈ I
h : ↑(((DFinsupp.evalAddMonoidHom max₂).comp (decomposeRingEquiv 𝒜).toRingHom.toAddMonoidHom) y) ∈ I
⊢ False