Theorems · Theorem · measure theory
intervalIntegral.integral_pos
∀ {f : ℝ → ℝ} {a b : ℝ},
a < b →
ContinuousOn f (Set.Icc a b) →
(∀ x ∈ Set.Ioc a b, 0 ≤ f x) → (∃ c ∈ Set.Icc a b, 0 < f c) → 0 < ∫ (x : ℝ) in a..b, f x- Cited by
- 0 results in Mathlib
- Foundations
- Depth 260 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- Set.Iccstatement and proof · cited by 1,702
- ContinuousOnstatement and proof · cited by 1,411
- MeasureTheory.MeasureSpace.volumestatement · cited by 1,323
- Set.Iocstatement and proof · cited by 971
- intervalIntegralstatement · cited by 546
- continuousOn_constproof · cited by 96
- intervalIntegral.integral_zeroproof · cited by 16
- LT.lt.trans_eq'proof · cited by 4
- intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_ltproof · cited by 2
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