Theorems · Inductive type · commutative algebra
Valuation.IsNontrivial
{R : Type u_3} →
{Γ₀ : Type u_4} → [inst : Ring R] → [inst_1 : LinearOrderedCommMonoidWithZero Γ₀] → Valuation R Γ₀ → PropA valuation on a ring is nontrivial if there exists an element with valuation
not equal to 0 or 1.
- Defined in
- Mathlib.RingTheory.Valuation.Basic
- Cited by
- 27 results in Mathlib
- Foundations
- Depth 2 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Ringstatement · cited by 7,463
- Valuationstatement · cited by 823
- LinearOrderedCommMonoidWithZerostatement · cited by 139
Cited by36
Results whose statement or proof uses this declaration.
- RatFunc.uniformizingPolynomialstatement and proof · cited by 7
- RatFunc.valuationIdealstatement and proof · cited by 5
- Valuation.IsNontrivial.exists_lt_onestatement and proof · cited by 4
- RatFunc.setOfPred_polynomial_valuation_lt_one_and_ne_zero_nonemptystatement and proof · cited by 4
- Valuation.IsNontrivial.casesOnstatement and proof · cited by 3
- RatFunc.uniformizingPolynomial_ne_zerostatement and proof · cited by 2
- RatFunc.valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_onestatement and proof · cited by 2
- Valuation.IsNontrivial.exists_val_nontrivialstatement and proof · cited by 2
- RatFunc.uniformizingPolynomial_isUniformizerstatement and proof · cited by 1
- RatFunc.valuation_isEquiv_valuationIdeal_adic_of_valuation_X_le_onestatement and proof · cited by 1
- RatFunc.valuation_uniformizingPolynomial_lt_onestatement and proof · cited by 1
- Valuation.IsNontrivial.exists_one_ltstatement and proof · cited by 1