Theorems · Theorem · order theory
Order.Ideal.isPrime_iff
∀ {P : Type u_1} [inst : Preorder P] (I : Order.Ideal P), I.IsPrime ↔ I.IsProper ∧ Order.IsPFilter (↑I)ᶜ- Defined in
- Mathlib.Order.PrimeIdeal
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses Quot.sound
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- SetLike.coestatement and proof · cited by 8,199
- Preorderstatement and proof · cited by 7,952
- Compl.complstatement and proof · cited by 2,925
- Order.Idealstatement and proof · cited by 102
- Order.Ideal.IsProperstatement and proof · cited by 23
- Order.Ideal.IsPrimestatement and proof · cited by 11
- Order.IsPFilterstatement and proof · cited by 6
- Order.Ideal.IsPrime.casesOnproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- Order.Ideal.IsPrime.of_mem_or_memproof · cited by 1