Theorems · Theorem · commutative algebra
Valuation.one_lt_val_iff
∀ {K : Type u_1} [inst : DivisionRing K] {Γ₀ : Type u_4} [inst_1 : LinearOrderedCommGroupWithZero Γ₀]
(v : Valuation K Γ₀) {x : K}, x ≠ 0 → (1 < v x ↔ v x⁻¹ < 1)- Defined in
- Mathlib.RingTheory.Valuation.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- DivisionRingstatement and proof · cited by 1,062
- Valuationstatement and proof · cited by 823
- LinearOrderedCommGroupWithZerostatement and proof · cited by 528
- map_inv₀proof · cited by 106
- inv_lt_one₀proof · cited by 9
- Valuation.pos_iffproof · cited by 5
Cited by4
Results whose statement or proof uses this declaration.
- Valuation.isEquiv_iff_val_lt_oneproof · cited by 3
- ValuationSubring.principalUnitGroup_le_principalUnitGroupproof · cited by 0
- Valuation.val_le_one_or_val_inv_lt_oneproof · cited by 0
- ValuationSubring.nonunits_le_nonunitsproof · cited by 0