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

tactic: apply Set.Subset.antisymm
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 : Set (ProjectiveSpectrum 𝒜)
⊢ zeroLocus 𝒜 ↑(vanishingIdeal t) = closure t
state_after: case h₁
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 : Set (ProjectiveSpectrum 𝒜)
⊢ zeroLocus 𝒜 ↑(vanishingIdeal t) ⊆ closure t

case h₂
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 : Set (ProjectiveSpectrum 𝒜)
⊢ closure t ⊆ zeroLocus 𝒜 ↑(vanishingIdeal t)

tactic: rintro x hx t' ⟨ht', ht⟩
state_before: case h₁
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 : Set (ProjectiveSpectrum 𝒜)
⊢ zeroLocus 𝒜 ↑(vanishingIdeal t) ⊆ closure t
state_after: case h₁.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 : Set (ProjectiveSpectrum 𝒜)
x : ProjectiveSpectrum 𝒜
hx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)
t' : Set (ProjectiveSpectrum 𝒜)
ht' : IsClosed t'
ht : t ⊆ t'
⊢ x ∈ t'

tactic: obtain ⟨fs, rfl⟩ : ∃ s, t' = zeroLocus 𝒜 s := by rwa [isClosed_iff_zeroLocus] at ht'
state_before: case h₁.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 : Set (ProjectiveSpectrum 𝒜)
x : ProjectiveSpectrum 𝒜
hx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)
t' : Set (ProjectiveSpectrum 𝒜)
ht' : IsClosed t'
ht : t ⊆ t'
⊢ x ∈ t'
state_after: case h₁.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 : Set (ProjectiveSpectrum 𝒜)
x : ProjectiveSpectrum 𝒜
hx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)
fs : Set A
ht' : IsClosed (zeroLocus 𝒜 fs)
ht : t ⊆ zeroLocus 𝒜 fs
⊢ x ∈ zeroLocus 𝒜 fs

tactic: rw [subset_zeroLocus_iff_subset_vanishingIdeal] at ht
state_before: case h₁.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 : Set (ProjectiveSpectrum 𝒜)
x : ProjectiveSpectrum 𝒜
hx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)
fs : Set A
ht' : IsClosed (zeroLocus 𝒜 fs)
ht : t ⊆ zeroLocus 𝒜 fs
⊢ x ∈ zeroLocus 𝒜 fs
state_after: case h₁.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 : Set (ProjectiveSpectrum 𝒜)
x : ProjectiveSpectrum 𝒜
hx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)
fs : Set A
ht' : IsClosed (zeroLocus 𝒜 fs)
ht : fs ⊆ ↑(vanishingIdeal t)
⊢ x ∈ zeroLocus 𝒜 fs

tactic: exact Set.Subset.trans ht hx
state_before: case h₁.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 : Set (ProjectiveSpectrum 𝒜)
x : ProjectiveSpectrum 𝒜
hx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)
fs : Set A
ht' : IsClosed (zeroLocus 𝒜 fs)
ht : fs ⊆ ↑(vanishingIdeal t)
⊢ x ∈ zeroLocus 𝒜 fs
state_after: no goals

tactic: rwa [isClosed_iff_zeroLocus] at ht'
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 : Set (ProjectiveSpectrum 𝒜)
x : ProjectiveSpectrum 𝒜
hx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)
t' : Set (ProjectiveSpectrum 𝒜)
ht' : IsClosed t'
ht : t ⊆ t'
⊢ ∃ s, t' = zeroLocus 𝒜 s
state_after: no goals

tactic: rw [(isClosed_zeroLocus _ _).closure_subset_iff]
state_before: case h₂
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 : Set (ProjectiveSpectrum 𝒜)
⊢ closure t ⊆ zeroLocus 𝒜 ↑(vanishingIdeal t)
state_after: case h₂
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 : Set (ProjectiveSpectrum 𝒜)
⊢ t ⊆ zeroLocus 𝒜 ↑(vanishingIdeal t)

tactic: exact subset_zeroLocus_vanishingIdeal 𝒜 t
state_before: case h₂
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 : Set (ProjectiveSpectrum 𝒜)
⊢ t ⊆ zeroLocus 𝒜 ↑(vanishingIdeal t)
state_after: no goals