Theorems · Theorem · field theory
Polynomial.zero_le_negOnePow_mul_eval_of_le_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
- 1 results in Mathlib
- Foundations
- Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- LT.lt.leproof · cited by 2,189
- MulZeroClass.mul_zeroproof · cited by 2,091
- le_reflproof · cited by 2,061
- Units.valstatement and proof · cited by 1,966
- Polynomial.natDegreestatement and proof · cited by 1,105
- Polynomial.evalstatement and proof · cited by 796
- Polynomial.leadingCoeffstatement and proof · cited by 498
- Int.negOnePowstatement and proof · cited by 156
- Polynomial.IsRootstatement and proof · cited by 152
- Polynomial.zero_lt_negOnePow_mul_eval_of_lt_roots_of_leadingCoeff_nonnegproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- Polynomial.negOnePow_mul_eval_le_zero_of_le_roots_of_leadingCoeff_nonposproof · cited by 0