ι : Type u_2
σ : Type u_3
R : Type ?u.119538
A : Type u_1
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I J : Ideal A
HI : ∃ S, I = span (Subtype.val '' S)
HJ : ∃ S, J = span (Subtype.val '' S)
⊢ ∃ S, I ⊔ J = span (Subtype.val '' S)
ι : Type u_2
σ : Type u_3
R : Type ?u.119538
A : Type u_1
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I J : Ideal A
HI : IsHomogeneous 𝒜 I
HJ : IsHomogeneous 𝒜 J
⊢ IsHomogeneous 𝒜 (I ⊔ J)
rw [iff_exists] at HI HJ⊢
case intro.intro
ι : Type u_2
σ : Type u_3
R : Type ?u.119538
A : Type u_1
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
s₁ s₂ : Set { x // x ∈ homogeneousSubmonoid 𝒜 }
⊢ ∃ S, span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' S)
ι : Type u_2
σ : Type u_3
R : Type ?u.119538
A : Type u_1
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I J : Ideal A
HI : ∃ S, I = span (Subtype.val '' S)
HJ : ∃ S, J = span (Subtype.val '' S)
⊢ ∃ S, I ⊔ J = span (Subtype.val '' S)
obtain ⟨⟨s₁, rfl⟩, ⟨s₂, rfl⟩⟩ := HI, HJ
case intro.intro
ι : Type u_2
σ : Type u_3
R : Type ?u.119538
A : Type u_1
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
s₁ s₂ : Set { x // x ∈ homogeneousSubmonoid 𝒜 }
⊢ span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' (s₁ ∪ s₂))
case intro.intro
ι : Type u_2
σ : Type u_3
R : Type ?u.119538
A : Type u_1
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
s₁ s₂ : Set { x // x ∈ homogeneousSubmonoid 𝒜 }
⊢ ∃ S, span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' S)
refine' ⟨s₁ ∪ s₂, _⟩
case intro.intro
ι : Type u_2
σ : Type u_3
R : Type ?u.119538
A : Type u_1
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
s₁ s₂ : Set { x // x ∈ homogeneousSubmonoid 𝒜 }
⊢ span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' s₁ ∪ Subtype.val '' s₂)
case intro.intro
ι : Type u_2
σ : Type u_3
R : Type ?u.119538
A : Type u_1
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
s₁ s₂ : Set { x // x ∈ homogeneousSubmonoid 𝒜 }
⊢ span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' (s₁ ∪ s₂))
rw [Set.image_union]
no goals
case intro.intro
ι : Type u_2
σ : Type u_3
R : Type ?u.119538
A : Type u_1
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
s₁ s₂ : Set { x // x ∈ homogeneousSubmonoid 𝒜 }
⊢ span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' s₁ ∪ Subtype.val '' s₂)
exact (Submodule.span_union _ _).symm