Theorem: Ideal.IsHomogeneous.sup
File: Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean

Proof steps:
state_before: ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
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)
state_after: ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
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)
tactic: rw [iff_exists] at HI HJ ⊢
state_before: ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
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)
state_after: case intro.intro
ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
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 ↥(homogeneousSubmonoid 𝒜)
⊢ ∃ S, span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' S)
tactic: obtain ⟨⟨s₁, rfl⟩, ⟨s₂, rfl⟩⟩ := HI, HJ
state_before: case intro.intro
ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
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 ↥(homogeneousSubmonoid 𝒜)
⊢ ∃ S, span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' S)
state_after: case intro.intro
ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
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 ↥(homogeneousSubmonoid 𝒜)
⊢ span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' (s₁ ∪ s₂))
tactic: refine ⟨s₁ ∪ s₂, ?_⟩
state_before: case intro.intro
ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
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 ↥(homogeneousSubmonoid 𝒜)
⊢ span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' (s₁ ∪ s₂))
state_after: case intro.intro
ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
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 ↥(homogeneousSubmonoid 𝒜)
⊢ span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' s₁ ∪ Subtype.val '' s₂)
tactic: rw [Set.image_union]
state_before: case intro.intro
ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
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 ↥(homogeneousSubmonoid 𝒜)
⊢ span (Subtype.val '' s₁) ⊔ span (Subtype.val '' s₂) = span (Subtype.val '' s₁ ∪ Subtype.val '' s₂)
state_after: no goals
tactic: exact (Submodule.span_union _ _).symm