Theorems · Theorem · commutative algebra

Valuation.Integers.isPrincipalIdealRing_iff_not_denselyOrdered

∀ {F : Type u_1} {Γ₀ : Type u_2} {O : Type u_3} [inst : Field F] [inst_1 : LinearOrderedCommGroupWithZero Γ₀]
  [inst_2 : CommRing O] [inst_3 : Algebra O F] {v : Valuation F Γ₀} [MulArchimedean ↥(MonoidHom.mrange v)],
  v.Integers O → (IsPrincipalIdealRing O ↔ ¬DenselyOrdered ↑(Set.range ⇑v))

Defined in: Mathlib.RingTheory.Valuation.Archimedean
Cited by: 1 results in Mathlib
Foundations: Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field LinearOrderedCommGroupWithZero CommRing Algebra MulArchimedean

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Setstatement · cited by 53,352
CommRingstatement and proof · cited by 17,173
Algebrastatement and proof · cited by 11,388
Fieldstatement and proof · cited by 7,404
Set.Elemstatement and proof · cited by 7,166
Algebra.algebraMapproof · cited by 4,706
Set.rangestatement and proof · cited by 4,705
Submonoidstatement · cited by 3,086
Unitsproof · cited by 2,804
Nontrivialproof · cited by 2,416
IsDomainproof · cited by 2,196

Cited by1

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Valuation.Integers.isPrincipalIdealRing_iff_not_denselyOrdered_mrangeproof · cited by 1