Theorems · Theorem · commutative algebra
Ideal.exists_not_mem_forall_mem_of_ne_of_liesOver
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)
[p.IsPrime] (q : Ideal S) [inst_4 : q.IsPrime] [q.LiesOver p] [Algebra.EssFiniteType R S] [Algebra.QuasiFiniteAt R q],
∃ s ∉ q, ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q → q'.LiesOver p → s ∈ q'- Defined in
- Mathlib.RingTheory.QuasiFinite.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites48
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- SetLike.coeproof · cited by 8,199
- Set.Elemproof · cited by 7,166
- Set.preimageproof · cited by 4,946
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapproof · cited by 4,706
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- TensorProduct.tmulproof · cited by 1,182
- map_oneproof · cited by 861
Cited by1
Results whose statement or proof uses this declaration.
- Algebra.exists_notMem_and_isIntegral_forall_mem_of_ne_of_liesOverproof · cited by 1