Theorems · Theorem · commutative algebra
IsLocalRing.of_unique_nonzero_prime
∀ {R : Type u_1} [inst : CommSemiring R], (∃! P, P ≠ ⊥ ∧ P.IsPrime) → IsLocalRing R- Defined in
- Mathlib.RingTheory.LocalRing.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 77 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.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- Top.topproof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.IsMaximalproof · cited by 452
- IsLocalRingstatement · cited by 339
- ExistsUniquestatement and proof · cited by 268
- ne_of_ltproof · cited by 203
- bot_lt_iff_ne_botproof · cited by 57
- Ideal.IsMaximal.isPrimeproof · cited by 53
- ne_bot_of_gtproof · cited by 18
Cited by1
Results whose statement or proof uses this declaration.
- IsDiscreteValuationRing.iff_pid_with_one_nonzero_primeproof · cited by 2