P. 255 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25401-25500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	minveclem3 25401*	Lemma for minvec 25408. The filter formed by taking elements successively closer to the infimum is Cauchy. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑋 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    ⇒   ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))
Theorem	minveclem4a 25402*	Lemma for minvec 25408. 𝐹 converges to a point 𝑃 in 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑋 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹))    ⇒   ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
Theorem	minveclem4b 25403*	Lemma for minvec 25408. The convergent point of the Cauchy sequence 𝐹 is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑋 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹))    ⇒   ⊢ (𝜑 → 𝑃 ∈ 𝑋)
Theorem	minveclem4 25404*	Lemma for minvec 25408. The convergent point of the Cauchy sequence 𝐹 attains the minimum distance, and so is closer to 𝐴 than any other point in 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
⊢ 𝑋 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹))    &   ⊢ 𝑇 = (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))
Theorem	minveclem5 25405*	Lemma for minvec 25408. Discharge the assumptions in minveclem4 25404. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑋 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))
Theorem	minveclem6 25406*	Lemma for minvec 25408. Any minimal point is less than 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)
⊢ 𝑋 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (((𝐴𝐷𝑥)↑2) ≤ ((𝑆↑2) + 0) ↔ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦))))
Theorem	minveclem7 25407*	Lemma for minvec 25408. Since any two minimal points are distance zero away from each other, the minimal point is unique. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑋 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ 𝐽 = (TopOpen‘𝑈)    &   ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))    &   ⊢ 𝑆 = inf(𝑅, ℝ, < )    &   ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))
Theorem	minvec 25408*	Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace 𝑊 that minimizes the distance to an arbitrary vector 𝐴 in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Proof shortened by AV, 3-Oct-2020.)
⊢ 𝑋 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑁 = (norm‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)    &   ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))    &   ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))
12.5.10  Projection Theorem
Theorem	pjthlem1 25409*	Lemma for pjth 25411. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 17-Oct-2015.) (Proof shortened by AV, 10-Jul-2022.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ ℂHil)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − 𝑥)))    &   ⊢ 𝑇 = ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))    ⇒   ⊢ (𝜑 → (𝐴 , 𝐵) = 0)
Theorem	pjthlem2 25410	Lemma for pjth 25411. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ ℂHil)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝑂 = (ocv‘𝑊)    &   ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐽))    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝑈 ⊕ (𝑂‘𝑈)))
Theorem	pjth 25411	Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed uniquely into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝑂 = (ocv‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑈 ⊕ (𝑂‘𝑈)) = 𝑉)
Theorem	pjth2 25412	Projection Theorem with abbreviations: A topologically closed subspace is a projection subspace. (Contributed by Mario Carneiro, 17-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ dom 𝐾)
Theorem	cldcss 25413	Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂHil → (𝑈 ∈ 𝐶 ↔ (𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽))))
Theorem	cldcss2 25414	Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂHil → 𝐶 = (𝐿 ∩ (Clsd‘𝐽)))
Theorem	hlhil 25415	Corollary of the Projection Theorem: A subcomplex Hilbert space is a Hilbert space (in the algebraic sense, meaning that all algebraically closed subspaces have a projection decomposition). (Contributed by Mario Carneiro, 17-Oct-2015.)
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Hil)
PART 13  BASIC REAL AND COMPLEX ANALYSIS
13.1  Continuity
Theorem	addcncf 25416*	The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝑋–cn→ℂ))
Theorem	subcncf 25417*	The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 − 𝐵)) ∈ (𝑋–cn→ℂ))
Theorem	mulcncf 25418*	The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.) Avoid ax-mulf 11220. (Revised by GG, 16-Mar-2025.)
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋–cn→ℂ))
Theorem	mulcncfOLD 25419*	Obsolete version of mulcncf 25418 as of 9-Apr-2025. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋–cn→ℂ))
Theorem	divcncf 25420*	The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→(ℂ ∖ {0})))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵)) ∈ (𝑋–cn→ℂ))
13.1.1  Intermediate value theorem
Theorem	pmltpclem1 25421*	Lemma for pmltpc 25423. (Contributed by Mario Carneiro, 1-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    &   ⊢ (𝜑 → (((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝐶) < (𝐹‘𝐵)) ∨ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝐶))))    ⇒   ⊢ (𝜑 → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))
Theorem	pmltpclem2 25422*	Lemma for pmltpc 25423. (Contributed by Mario Carneiro, 1-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (ℝ ↑pm ℝ))    &   ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑈 ≤ 𝑉)    &   ⊢ (𝜑 → 𝑊 ≤ 𝑋)    &   ⊢ (𝜑 → ¬ (𝐹‘𝑈) ≤ (𝐹‘𝑉))    &   ⊢ (𝜑 → ¬ (𝐹‘𝑋) ≤ (𝐹‘𝑊))    ⇒   ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))
Theorem	pmltpc 25423*	Any function on the reals is either increasing, decreasing, or has a triple of points in a vee formation. (This theorem was created on demand by Mario Carneiro for the 6PCM conference in Bialystok, 1-Jul-2014.) (Contributed by Mario Carneiro, 1-Jul-2014.)
⊢ ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ∨ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)) ∨ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))))
Theorem	ivthlem1 25424*	Lemma for ivth 25427. The set 𝑆 of all 𝑥 values with (𝐹‘𝑥) less than 𝑈 is lower bounded by 𝐴 and upper bounded by 𝐵. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈}    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵))
Theorem	ivthlem2 25425*	Lemma for ivth 25427. Show that the supremum of 𝑆 cannot be less than 𝑈. If it was, continuity of 𝐹 implies that there are points just above the supremum that are also less than 𝑈, a contradiction. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈}    &   ⊢ 𝐶 = sup(𝑆, ℝ, < )    ⇒   ⊢ (𝜑 → ¬ (𝐹‘𝐶) < 𝑈)
Theorem	ivthlem3 25426*	Lemma for ivth 25427, the intermediate value theorem. Show that (𝐹‘𝐶) cannot be greater than 𝑈, and so establish the existence of a root of the function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 17-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    &   ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈}    &   ⊢ 𝐶 = sup(𝑆, ℝ, < )    ⇒   ⊢ (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ∧ (𝐹‘𝐶) = 𝑈))
Theorem	ivth 25427*	The intermediate value theorem, increasing case. This is Metamath 100 proof #79. (Contributed by Paul Chapman, 22-Jan-2008.) (Proof shortened by Mario Carneiro, 30-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
Theorem	ivth2 25428*	The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   ⊢ (𝜑 → ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴)))    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)
Theorem	ivthle 25429*	The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   ⊢ (𝜑 → ((𝐹‘𝐴) ≤ 𝑈 ∧ 𝑈 ≤ (𝐹‘𝐵)))    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
Theorem	ivthle2 25430*	The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    &   ⊢ (𝜑 → ((𝐹‘𝐵) ≤ 𝑈 ∧ 𝑈 ≤ (𝐹‘𝐴)))    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹‘𝑐) = 𝑈)
Theorem	ivthicc 25431*	The interval between any two points of a continuous real function is contained in the range of the function. Equivalently, the range of a continuous real function is convex. (Contributed by Mario Carneiro, 12-Aug-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → 𝑁 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐹‘𝑀)[,](𝐹‘𝑁)) ⊆ ran 𝐹)
Theorem	evthicc 25432*	Specialization of the Extreme Value Theorem to a closed interval of ℝ. (Contributed by Mario Carneiro, 12-Aug-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤)))
Theorem	evthicc2 25433*	Combine ivthicc 25431 with evthicc 25432 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran 𝐹 = (𝑥[,]𝑦))
Theorem	cniccbdd 25434*	A continuous function on a closed interval is bounded. (Contributed by Mario Carneiro, 7-Sep-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑦)) ≤ 𝑥)
13.2  Integrals
13.2.1  Lebesgue measure
Syntax	covol 25435	Extend class notation with the outer Lebesgue measure.
class vol*
Syntax	cvol 25436	Extend class notation with the Lebesgue measure.
class vol
Definition	df-ovol 25437*	Define the outer Lebesgue measure for subsets of the reals. Here 𝑓 is a function from the positive integers to pairs ⟨𝑎, 𝑏⟩ with 𝑎 ≤ 𝑏, and the outer volume of the set 𝑥 is the infimum over all such functions such that the union of the open intervals (𝑎, 𝑏) covers 𝑥 of the sum of 𝑏 − 𝑎. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)
⊢ vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
Definition	df-vol 25438*	Define the Lebesgue measure, which is just the outer measure with a peculiar domain of definition. The property of being Lebesgue-measurable can be expressed as 𝐴 ∈ dom vol. (Contributed by Mario Carneiro, 17-Mar-2014.)
⊢ vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})
Theorem	ovolfcl 25439	Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑁)) ≤ (2nd ‘(𝐹‘𝑁))))
Theorem	ovolfioo 25440*	Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑛)))))
Theorem	ovolficc 25441*	Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ([,] ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2nd ‘(𝐹‘𝑛)))))
Theorem	ovolficcss 25442	Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
Theorem	ovolfsval 25443	The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹)    ⇒   ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁))))
Theorem	ovolfsf 25444	Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹)    ⇒   ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞))
Theorem	ovolsf 25445	Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹)    &   ⊢ 𝑆 = seq1( + , 𝐺)    ⇒   ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
Theorem	ovolval 25446*	The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)
⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    ⇒   ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Theorem	elovolmlem 25447	Lemma for elovolm 25448 and related theorems. (Contributed by BJ, 23-Jul-2022.)
⊢ (𝐹 ∈ ((𝐴 ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶(𝐴 ∩ (ℝ × ℝ)))
Theorem	elovolm 25448*	Elementhood in the set 𝑀 of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    ⇒   ⊢ (𝐵 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))
Theorem	elovolmr 25449*	Sufficient condition for elementhood in the set 𝑀. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    ⇒   ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)
Theorem	ovolmge0 25450*	The set 𝑀 is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    ⇒   ⊢ (𝐵 ∈ 𝑀 → 0 ≤ 𝐵)
Theorem	ovolcl 25451	The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
Theorem	ovollb 25452	The outer volume is a lower bound on the sum of all interval coverings of 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    ⇒   ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Theorem	ovolgelb 25453*	The outer volume is the greatest lower bound on the sum of all interval coverings of 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑔))    ⇒   ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵)))
Theorem	ovolge0 25454	The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
Theorem	ovolf 25455	The domain and codomain of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
⊢ vol*:𝒫 ℝ⟶(0[,]+∞)
Theorem	ovollecl 25456	If an outer volume is bounded above, then it is real. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ (vol*‘𝐴) ≤ 𝐵) → (vol*‘𝐴) ∈ ℝ)
Theorem	ovolsslem 25457*	Lemma for ovolss 25458. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    &   ⊢ 𝑁 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))
Theorem	ovolss 25458	The volume of a set is monotone with respect to set inclusion. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))
Theorem	ovolsscl 25459	If a set is contained in another of bounded measure, it too is bounded. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐴) ∈ ℝ)
Theorem	ovolssnul 25460	A subset of a nullset is null. (Contributed by Mario Carneiro, 19-Mar-2014.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) = 0)
Theorem	ovollb2lem 25461*	Lemma for ovollb2 25462. (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩)    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹))    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)    ⇒   ⊢ (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
Theorem	ovollb2 25462	It is often more convenient to do calculations with *closed* coverings rather than open ones; here we show that it makes no difference (compare ovollb 25452). (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    ⇒   ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Theorem	ovolctb 25463	The volume of a denumerable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≈ ℕ) → (vol*‘𝐴) = 0)
Theorem	ovolq 25464	The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
⊢ (vol*‘ℚ) = 0
Theorem	ovolctb2 25465	The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘𝐴) = 0)
Theorem	ovol0 25466	The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
⊢ (vol*‘∅) = 0
Theorem	ovolfi 25467	A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ ℝ) → (vol*‘𝐴) = 0)
Theorem	ovolsn 25468	A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)
⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0)
Theorem	ovolunlem1a 25469*	Lemma for ovolun 25472. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))    &   ⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))    &   ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))    &   ⊢ (𝜑 → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))    &   ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺))    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))))    ⇒   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
Theorem	ovolunlem1 25470*	Lemma for ovolun 25472. (Contributed by Mario Carneiro, 12-Jun-2014.)
⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))    &   ⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))    &   ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))    &   ⊢ (𝜑 → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))    &   ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺))    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))))    ⇒   ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
Theorem	ovolunlem2 25471	Lemma for ovolun 25472. (Contributed by Mario Carneiro, 12-Jun-2014.)
⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))    &   ⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
Theorem	ovolun 25472	The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 25478, this does not require any choice principles.) (Contributed by Mario Carneiro, 12-Jun-2014.)
⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
Theorem	ovolunnul 25473	Adding a nullset does not change the measure of a set. (Contributed by Mario Carneiro, 25-Mar-2015.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘(𝐴 ∪ 𝐵)) = (vol*‘𝐴))
Theorem	ovolfiniun 25474*	The Lebesgue outer measure function is finitely sub-additive. Finite sum version. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵))
Theorem	ovoliunlem1 25475*	Lemma for ovoliun 25478. (Contributed by Mario Carneiro, 12-Jun-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ 𝑇 = seq1( + , 𝐺)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛)))    &   ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   ⊢ 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽‘𝑘)))‘(2nd ‘(𝐽‘𝑘))))    &   ⊢ (𝜑 → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))    &   ⊢ (𝜑 → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝐿 ∈ ℤ)    &   ⊢ (𝜑 → ∀𝑤 ∈ (1...𝐾)(1st ‘(𝐽‘𝑤)) ≤ 𝐿)    ⇒   ⊢ (𝜑 → (𝑈‘𝐾) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Theorem	ovoliunlem2 25476*	Lemma for ovoliun 25478. (Contributed by Mario Carneiro, 12-Jun-2014.)
⊢ 𝑇 = seq1( + , 𝐺)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛)))    &   ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))    &   ⊢ 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽‘𝑘)))‘(2nd ‘(𝐽‘𝑘))))    &   ⊢ (𝜑 → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))    &   ⊢ (𝜑 → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))    ⇒   ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Theorem	ovoliunlem3 25477*	Lemma for ovoliun 25478. (Contributed by Mario Carneiro, 12-Jun-2014.)
⊢ 𝑇 = seq1( + , 𝐺)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    &   ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Theorem	ovoliun 25478*	The Lebesgue outer measure function is countably sub-additive. (Many books allow +∞ as a value for one of the sets in the sum, but in our setup we can't do arithmetic on infinity, and in any case the volume of a union containing an infinitely large set is already infinitely large by monotonicity ovolss 25458, so we need not consider this case here, although we do allow the sum itself to be infinite.) (Contributed by Mario Carneiro, 12-Jun-2014.)
⊢ 𝑇 = seq1( + , 𝐺)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    ⇒   ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
Theorem	ovoliun2 25479*	The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 25478.) (Contributed by Mario Carneiro, 12-Jun-2014.)
⊢ 𝑇 = seq1( + , 𝐺)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)    &   ⊢ (𝜑 → 𝑇 ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))
Theorem	ovoliunnul 25480*	A countable union of nullsets is null. (Contributed by Mario Carneiro, 8-Apr-2015.)
⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0)
Theorem	shft2rab 25481*	If 𝐵 is a shift of 𝐴 by 𝐶, then 𝐴 is a shift of 𝐵 by -𝐶. (Contributed by Mario Carneiro, 22-Mar-2014.) (Revised by Mario Carneiro, 6-Apr-2015.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴})    ⇒   ⊢ (𝜑 → 𝐴 = {𝑦 ∈ ℝ ∣ (𝑦 − -𝐶) ∈ 𝐵})
Theorem	ovolshftlem1 25482*	Lemma for ovolshft 25484. (Contributed by Mario Carneiro, 22-Mar-2014.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴})    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩)    &   ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))    ⇒   ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)
Theorem	ovolshftlem2 25483*	Lemma for ovolshft 25484. (Contributed by Mario Carneiro, 22-Mar-2014.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴})    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    ⇒   ⊢ (𝜑 → {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ⊆ 𝑀)
Theorem	ovolshft 25484*	The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴})    ⇒   ⊢ (𝜑 → (vol*‘𝐴) = (vol*‘𝐵))
Theorem	sca2rab 25485*	If 𝐵 is a scale of 𝐴 by 𝐶, then 𝐴 is a scale of 𝐵 by 1 / 𝐶. (Contributed by Mario Carneiro, 22-Mar-2014.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})    ⇒   ⊢ (𝜑 → 𝐴 = {𝑦 ∈ ℝ ∣ ((1 / 𝐶) · 𝑦) ∈ 𝐵})
Theorem	ovolscalem1 25486*	Lemma for ovolsca 25488. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})    &   ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩)    &   ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))    ⇒   ⊢ (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
Theorem	ovolscalem2 25487*	Lemma for ovolshft 25484. (Contributed by Mario Carneiro, 22-Mar-2014.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})    &   ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)    ⇒   ⊢ (𝜑 → (vol*‘𝐵) ≤ ((vol*‘𝐴) / 𝐶))
Theorem	ovolsca 25488*	The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})    &   ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)    ⇒   ⊢ (𝜑 → (vol*‘𝐵) = ((vol*‘𝐴) / 𝐶))
Theorem	ovolicc1 25489*	The measure of a closed interval is lower bounded by its length. (Contributed by Mario Carneiro, 13-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))    ⇒   ⊢ (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵 − 𝐴))
Theorem	ovolicc2lem1 25490*	Lemma for ovolicc2 25495. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)    &   ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ 𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2nd ‘(𝐹‘(𝐺‘𝑋))))))
Theorem	ovolicc2lem2 25491*	Lemma for ovolicc2 25495. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)    &   ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)    &   ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}    &   ⊢ (𝜑 → 𝐻:𝑇⟶𝑇)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))    &   ⊢ 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾‘𝑛)}    ⇒   ⊢ ((𝜑 ∧ (𝑁 ∈ ℕ ∧ ¬ 𝑁 ∈ 𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑁)))) ≤ 𝐵)
Theorem	ovolicc2lem3 25492*	Lemma for ovolicc2 25495. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)    &   ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)    &   ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}    &   ⊢ (𝜑 → 𝐻:𝑇⟶𝑇)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))    &   ⊢ 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾‘𝑛)}    ⇒   ⊢ ((𝜑 ∧ (𝑁 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑃 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})) → (𝑁 = 𝑃 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑁)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑃))))))
Theorem	ovolicc2lem4 25493*	Lemma for ovolicc2 25495. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by AV, 17-Sep-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)    &   ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)    &   ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}    &   ⊢ (𝜑 → 𝐻:𝑇⟶𝑇)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))    &   ⊢ 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾‘𝑛)}    &   ⊢ 𝑀 = inf(𝑊, ℝ, < )    ⇒   ⊢ (𝜑 → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Theorem	ovolicc2lem5 25494*	Lemma for ovolicc2 25495. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)    &   ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)    &   ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}    ⇒   ⊢ (𝜑 → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Theorem	ovolicc2 25495*	The measure of a closed interval is upper bounded by its length. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}    ⇒   ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol*‘(𝐴[,]𝐵)))
Theorem	ovolicc 25496	The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵 − 𝐴))
Theorem	ovolicopnf 25497	The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝐴 ∈ ℝ → (vol*‘(𝐴[,)+∞)) = +∞)
Theorem	ovolre 25498	The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (vol*‘ℝ) = +∞
Theorem	ismbl 25499*	The predicate "𝐴 is Lebesgue-measurable". A set is measurable if it splits every other set 𝑥 in a "nice" way, that is, if the measure of the pieces 𝑥 ∩ 𝐴 and 𝑥 ∖ 𝐴 sum up to the measure of 𝑥 (assuming that the measure of 𝑥 is a real number, so that this addition makes sense). (Contributed by Mario Carneiro, 17-Mar-2014.)
⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))))
Theorem	ismbl2 25500*	From ovolun 25472, it suffices to show that the measure of 𝑥 is at least the sum of the measures of 𝑥 ∩ 𝐴 and 𝑥 ∖ 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))