Theorems · Theorem · field theory
Polynomial.eval_lt_zero_of_roots_lt_of_leadingCoeff_nonpos
∀ {P : Polynomial ℝ} {x : ℝ}, (∀ (y : ℝ), P.IsRoot y → y < x) → P.leadingCoeff ≤ 0 → Polynomial.eval x P < 0- Defined in
- Mathlib.Analysis.Polynomial.Order
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Polynomialstatement and proof · cited by 5,681
- Polynomial.evalstatement and proof · cited by 796
- neg_zeroproof · cited by 542
- Polynomial.leadingCoeffstatement and proof · cited by 498
- Polynomial.IsRootstatement and proof · cited by 152
- neg_posproof · cited by 74
- neg_eq_iff_eq_negproof · cited by 60
- Polynomial.eval_negproof · cited by 30
- le_negproof · cited by 16
- Polynomial.leadingCoeff_negproof · cited by 10
- Polynomial.zero_lt_eval_of_roots_lt_of_leadingCoeff_nonnegproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- Polynomial.zero_lt_negOnePow_mul_eval_of_lt_roots_of_leadingCoeff_nonnegproof · cited by 2