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
x y : ProjectiveSpectrum 𝒜
⊢ x.asHomogeneousIdeal ≤ y.asHomogeneousIdeal ↔ ↑x.asHomogeneousIdeal ⊆ ↑y.asHomogeneousIdeal
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
x y : ProjectiveSpectrum 𝒜
⊢ x ≤ y ↔ y ∈ closure {x}
rw [← as_ideal_le_as_ideal, ← zeroLocus_vanishingIdeal_eq_closure, mem_zeroLocus,
  vanishingIdeal_singleton]
no goals
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
x y : ProjectiveSpectrum 𝒜
⊢ x.asHomogeneousIdeal ≤ y.asHomogeneousIdeal ↔ ↑x.asHomogeneousIdeal ⊆ ↑y.asHomogeneousIdeal
simp only [as_ideal_le_as_ideal, coe_subset_coe]