integral_pos — Mathlib

English

If Ι(a,b) ⊆ Ι(c,d), then |∫_a^b f| ≤ |∫_c^d f| for integrable f.

Русский

Если unordered-интервал I(a,b) ⊆ I(c,d), то |∫_a^b f| ≤ |∫_c^d f|.

LaTeX

$$$Ι(a,b) \\subseteq Ι(c,d) \\Rightarrow |∫_a^b f| ≤ |∫_c^d f|$.$$

Lean4

theorem integral_pos (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hle : ∀ x ∈ Ioc a b, 0 ≤ f x)
    (hlt : ∃ c ∈ Icc a b, 0 < f c) : 0 < ∫ x in a..b, f x :=
  (integral_lt_integral_of_continuousOn_of_le_of_exists_lt hab continuousOn_const hfc hle hlt).trans_eq' (by simp)

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