Theorems · Theorem · algebraic geometry
PrimeSpectrum.exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopen
∀ {R : Type u} [inst : CommSemiring R] {s : Set (PrimeSpectrum R)},
IsClopen s → ∃ e f, e * f = 0 ∧ e + f = 1 ∧ s = ↑(PrimeSpectrum.basicOpen e) ∧ sᶜ = ↑(PrimeSpectrum.basicOpen f)- Cited by
- 2 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites59
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- CommSemiringstatement and proof · cited by 10,911
- Top.topproof · cited by 9,680
- SetLike.coestatement and proof · cited by 8,199
- Set.imageproof · cited by 5,609
- Idealproof · cited by 4,748
- Bot.botproof · cited by 4,720
- Set.univproof · cited by 3,945
- Compl.complstatement and proof · cited by 2,925
- Nontrivialproof · cited by 2,416
- TopologicalSpace.Opensstatement · cited by 2,040
- pow_zeroproof · cited by 1,094
Cited by2
Results whose statement or proof uses this declaration.
- PrimeSpectrum.exists_idempotent_basicOpen_eq_of_isClopenproof · cited by 2
- PrimeSpectrum.isClopen_iff_mul_addproof · cited by 1