Theorems · Theorem · commutative algebra
Valuation.zero_iff
∀ {K : Type u_1} [inst : DivisionRing K] {Γ₀ : Type u_4} [inst_1 : LinearOrderedCommMonoidWithZero Γ₀] [Nontrivial Γ₀]
(v : Valuation K Γ₀) {x : K}, v x = 0 ↔ x = 0If v is a valuation on a division ring then v(x) = 0 iff x = 0.
- Defined in
- Mathlib.RingTheory.Valuation.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 45 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Nontrivialstatement and proof · cited by 2,416
- DivisionRingstatement and proof · cited by 1,062
- Valuationstatement and proof · cited by 823
- LinearOrderedCommMonoidWithZerostatement and proof · cited by 139
- map_eq_zeroproof · cited by 10
Cited by9
Results whose statement or proof uses this declaration.
- Valuation.Integers.dvd_of_leproof · cited by 4
- Valuation.exists_pow_Uniformizerproof · cited by 3
- Valuation.isEquiv_iff_val_lt_oneproof · cited by 3
- Valuation.RankOne.exists_val_ltproof · cited by 1
- Valuation.inversion_estimateproof · cited by 1
- Valued.closure_coe_completion_v_ltproof · cited by 1
- Valued.extension_eq_zero_iffproof · cited by 0
- AddValuation.top_iffproof · cited by 0
- Valuation.pow_coeff_zero_ne_zero_of_unitproof · cited by 0