Theorems · Theorem · commutative algebra
IsLocalization.sec_snd_ne_zero
∀ {R : Type u_1} [inst : CommSemiring R] {M : Submonoid R} {S : Type u_2} [inst_1 : CommSemiring S]
[inst_2 : Algebra R S] [inst_3 : IsLocalization M S] [Nontrivial R],
M ≤ nonZeroDivisors R → ∀ (x : S), ↑(IsLocalization.sec M x).2 ≠ 0- Defined in
- Mathlib.RingTheory.Localization.Defs
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 24 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Submonoidstatement and proof · cited by 3,086
- Nontrivialstatement and proof · cited by 2,416
- nonZeroDivisorsstatement and proof · cited by 895
- IsLocalizationstatement and proof · cited by 636
- IsLocalization.secstatement and proof · cited by 18
- nonZeroDivisors.coe_ne_zeroproof · cited by 17
Cited by1
Results whose statement or proof uses this declaration.
- ClassGroup.mk_eq_mk_of_coe_idealproof · cited by 2