Theorems · Theorem · field theory
Polynomial.zero_lt_negOnePow_mul_eval_of_lt_roots_of_leadingCoeff_nonneg
∀ {P : Polynomial ℝ} {x : ℝ},
(∀ (y : ℝ), P.IsRoot y → x < y) → 0 ≤ P.leadingCoeff → 0 < ↑↑(↑P.natDegree).negOnePow * Polynomial.eval x P- Defined in
- Mathlib.Analysis.Polynomial.Order
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites37
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
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- Unitsproof · cited by 2,804
- Nat.cast_oneproof · cited by 2,501
- mul_commproof · cited by 2,262
- Units.valstatement and proof · cited by 1,966
- mul_assocproof · cited by 1,667
- Polynomial.Xproof · cited by 1,639
- Polynomial.natDegreestatement and proof · cited by 1,105
- neg_negproof · cited by 960
Cited by2
Results whose statement or proof uses this declaration.
- Polynomial.zero_le_negOnePow_mul_eval_of_le_roots_of_leadingCoeff_nonnegproof · cited by 1
- Polynomial.negOnePow_mul_eval_lt_zero_of_lt_roots_of_leadingCoeff_nonposproof · cited by 0