Theorems · Theorem · algebraic geometry
PrimeSpectrum.isClopen_iff_mul_add
∀ {R : Type u} [inst : CommSemiring R] {s : Set (PrimeSpectrum R)},
IsClopen s ↔ ∃ e f, e * f = 0 ∧ e + f = 1 ∧ s = ↑(PrimeSpectrum.basicOpen e)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 89 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.
Cites10
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
- SetLike.coestatement and proof · cited by 8,199
- Compl.complproof · cited by 2,925
- TopologicalSpace.Opensstatement · cited by 2,040
- PrimeSpectrumstatement and proof · cited by 625
- IsClopenstatement and proof · cited by 189
- PrimeSpectrum.basicOpenstatement and proof · cited by 163
- PrimeSpectrum.exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopenproof · cited by 2
- PrimeSpectrum.isClopen_basicOpen_of_mul_addproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- PrimeSpectrum.isClopen_iff_mul_add_zeroLocusproof · cited by 0