Theorem: ProjectiveSpectrum.sup_vanishingIdeal_le
File: Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
t t' : Set (ProjectiveSpectrum 𝒜)
⊢ vanishingIdeal t ⊔ vanishingIdeal t' ≤ vanishingIdeal (t ∩ t')
state_after: R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
t t' : Set (ProjectiveSpectrum 𝒜)
r : A
⊢ r ∈ vanishingIdeal t ⊔ vanishingIdeal t' → r ∈ vanishingIdeal (t ∩ t')
tactic: intro r
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
t t' : Set (ProjectiveSpectrum 𝒜)
r : A
⊢ r ∈ vanishingIdeal t ⊔ vanishingIdeal t' → r ∈ vanishingIdeal (t ∩ t')
state_after: R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
t t' : Set (ProjectiveSpectrum 𝒜)
r : A
⊢ (∃ y ∈ (vanishingIdeal t).toIdeal, ∃ z ∈ (vanishingIdeal t').toIdeal, y + z = r) →
    ∀ x ∈ t ∩ t', r ∈ x.asHomogeneousIdeal
tactic: rw [← HomogeneousIdeal.mem_iff, HomogeneousIdeal.toIdeal_sup, mem_vanishingIdeal,
    Submodule.mem_sup]
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
t t' : Set (ProjectiveSpectrum 𝒜)
r : A
⊢ (∃ y ∈ (vanishingIdeal t).toIdeal, ∃ z ∈ (vanishingIdeal t').toIdeal, y + z = r) →
    ∀ x ∈ t ∩ t', r ∈ x.asHomogeneousIdeal
state_after: case intro.intro.intro.intro.intro
R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
t t' : Set (ProjectiveSpectrum 𝒜)
f : A
hf : f ∈ (vanishingIdeal t).toIdeal
g : A
hg : g ∈ (vanishingIdeal t').toIdeal
x : ProjectiveSpectrum 𝒜
hxt : x ∈ t
hxt' : x ∈ t'
⊢ f + g ∈ x.asHomogeneousIdeal
tactic: rintro ⟨f, hf, g, hg, rfl⟩ x ⟨hxt, hxt'⟩
state_before: case intro.intro.intro.intro.intro
R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
t t' : Set (ProjectiveSpectrum 𝒜)
f : A
hf : f ∈ (vanishingIdeal t).toIdeal
g : A
hg : g ∈ (vanishingIdeal t').toIdeal
x : ProjectiveSpectrum 𝒜
hxt : x ∈ t
hxt' : x ∈ t'
⊢ f + g ∈ x.asHomogeneousIdeal
state_after: case intro.intro.intro.intro.intro
R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
t t' : Set (ProjectiveSpectrum 𝒜)
f : A
hf : ∀ x ∈ t, f ∈ x.asHomogeneousIdeal
g : A
hg : ∀ x ∈ t', g ∈ x.asHomogeneousIdeal
x : ProjectiveSpectrum 𝒜
hxt : x ∈ t
hxt' : x ∈ t'
⊢ f + g ∈ x.asHomogeneousIdeal
tactic: erw [mem_vanishingIdeal] at hf hg
state_before: case intro.intro.intro.intro.intro
R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
t t' : Set (ProjectiveSpectrum 𝒜)
f : A
hf : ∀ x ∈ t, f ∈ x.asHomogeneousIdeal
g : A
hg : ∀ x ∈ t', g ∈ x.asHomogeneousIdeal
x : ProjectiveSpectrum 𝒜
hxt : x ∈ t
hxt' : x ∈ t'
⊢ f + g ∈ x.asHomogeneousIdeal
state_after: no goals
tactic: apply Submodule.add_mem <;> solve_by_elim