Theorems · Theorem · commutative algebra
Ring.not_isField_of_ne_of_ne
∀ {R : Type u_5} [inst : CommSemiring R] [Nontrivial R] {I : Ideal R}, I ≠ ⊥ → I ≠ ⊤ → ¬IsField R- Defined in
- Mathlib.RingTheory.Ideal.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringNontrivial
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.
- CommSemiringstatement and proof · cited by 10,911
- Top.topstatement and proof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- Nontrivialstatement and proof · cited by 2,416
- bot_leproof · cited by 306
- IsFieldstatement and proof · cited by 103
- Ideal.IsMaximal.eq_of_leproof · cited by 39
- Ring.isField_iff_maximal_botproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- Ring.not_isField_iff_exists_primeproof · cited by 4
- Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_topproof · cited by 3