P. 238 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23701-23800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rrxf 23701*	Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝐹:𝐼⟶ℝ)
Theorem	rrxfsupp 23702*	Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin)
Theorem	rrxsuppss 23703*	Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹 supp 0) ⊆ 𝐼)
Theorem	rrxmvallem 23704*	Support of the function used for building the distance . (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝑘 ∈ 𝐼 ↦ (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0)))
Theorem	rrxmval 23705*	The value of the Euclidean metric. Compare with rrnmval 34526. (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
Theorem	rrxmfval 23706*	The value of the Euclidean metric. Compare with rrnval 34525. (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
Theorem	rrxmetlem 23707*	Lemma for rrxmet 23708. (Contributed by Thierry Arnoux, 5-Jul-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐴)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = Σ𝑘 ∈ 𝐴 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
Theorem	rrxmet 23708*	Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.) (Revised by Thierry Arnoux, 30-Jun-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))
Theorem	rrxdstprj1 23709*	The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 7-Jul-2019.)
⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    &   ⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))
Theorem	rrxbasefi 23710	The base of the generalized real Euclidean space, when the dimension of the space is finite. This justifies the use of (ℝ ↑𝑚 𝑋) for the development of the Lebesgue measure theory for n-dimensional real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝐻 = (ℝ^‘𝑋)    &   ⊢ 𝐵 = (Base‘𝐻)    ⇒   ⊢ (𝜑 → 𝐵 = (ℝ ↑𝑚 𝑋))
Theorem	rrxdsfi 23711*	The distance over generalized Euclidean spaces. Finite dimensional case. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐻 = (ℝ^‘𝐼)    &   ⊢ 𝐵 = (ℝ ↑𝑚 𝐼)    ⇒   ⊢ (𝐼 ∈ Fin → (dist‘𝐻) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
Theorem	rrxmetfi 23712	Euclidean space is a metric space. Finite dimensional version. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    ⇒   ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑𝑚 𝐼)))
Theorem	rrxdsfival 23713*	The value of the Euclidean distance function in a generalized real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.)
⊢ 𝑋 = (ℝ ↑𝑚 𝐼)    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    ⇒   ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
Theorem	ehlval 23714	Value of the Euclidean space of dimension 𝑁. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐸 = (𝔼hil‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁)))
Theorem	ehlbase 23715	The base of the Euclidean space is the set of n-tuples of real numbers. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐸 = (𝔼hil‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑𝑚 (1...𝑁)) = (Base‘𝐸))
Theorem	ehl0base 23716	The base of the Euclidean space of dimension 0 consists only of one element, the empty set. (Contributed by AV, 12-Feb-2023.)
⊢ 𝐸 = (𝔼hil‘0)    ⇒   ⊢ (Base‘𝐸) = {∅}
Theorem	ehl0 23717	The Euclidean space of dimension 0 consists of the neutral element only. (Contributed by AV, 12-Feb-2023.)
⊢ 𝐸 = (𝔼hil‘0)    &   ⊢ 0 = (0g‘𝐸)    ⇒   ⊢ (Base‘𝐸) = { 0 }
Theorem	ehleudis 23718*	The Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.)
⊢ 𝐼 = (1...𝑁)    &   ⊢ 𝐸 = (𝔼hil‘𝑁)    &   ⊢ 𝑋 = (ℝ ↑𝑚 𝐼)    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
Theorem	ehleudisval 23719*	The value of the Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.)
⊢ 𝐼 = (1...𝑁)    &   ⊢ 𝐸 = (𝔼hil‘𝑁)    &   ⊢ 𝑋 = (ℝ ↑𝑚 𝐼)    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
Theorem	ehl1eudis 23720*	The Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐸 = (𝔼hil‘1)    &   ⊢ 𝑋 = (ℝ ↑𝑚 {1})    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1))))
Theorem	ehl1eudisval 23721	The value of the Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐸 = (𝔼hil‘1)    &   ⊢ 𝑋 = (ℝ ↑𝑚 {1})    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (abs‘((𝐹‘1) − (𝐺‘1))))
Theorem	ehl2eudis 23722*	The Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐸 = (𝔼hil‘2)    &   ⊢ 𝑋 = (ℝ ↑𝑚 {1, 2})    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))))
Theorem	ehl2eudisval 23723	The value of the Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐸 = (𝔼hil‘2)    &   ⊢ 𝑋 = (ℝ ↑𝑚 {1, 2})    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘((((𝐹‘1) − (𝐺‘1))↑2) + (((𝐹‘2) − (𝐺‘2))↑2))))
12.5.9  Minimizing Vector Theorem
Theorem	minveclem1 23724*	Lemma for minvec 23736. The set of all distances from points of 𝑌 to 𝐴 are a nonempty set of nonnegative reals. (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)))    ⇒   ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
Theorem	minveclem4c 23725*	Lemma for minvec 23736. The infimum of the distances to 𝐴 is a real number. (Contributed by Mario Carneiro, 16-Jun-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, < )    ⇒   ⊢ (𝜑 → 𝑆 ∈ ℝ)
Theorem	minveclem2 23726*	Lemma for minvec 23736. Any two points 𝐾 and 𝐿 in 𝑌 are close to each other if they are close to the infimum of distance to 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 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑌)    &   ⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵))    &   ⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵))    ⇒   ⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵))
Theorem	minveclem3a 23727*	Lemma for minvec 23736. 𝐷 is a complete metric when restricted to 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))    ⇒   ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
Theorem	minveclem3b 23728*	Lemma for minvec 23736. The set of vectors within a fixed distance of the infimum forms a filter base. (Contributed 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)})    ⇒   ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))
Theorem	minveclem3 23729*	Lemma for minvec 23736. 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 23730*	Lemma for minvec 23736. 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 23731*	Lemma for minvec 23736. 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 23732*	Lemma for minvec 23736. 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 23733*	Lemma for minvec 23736. Discharge the assumptions in minveclem4 23732. (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 23734*	Lemma for minvec 23736. 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 23735*	Lemma for minvec 23736. 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 23736*	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 23737*	Lemma for pjth 23739. (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 23738	Lemma for pjth 23739. (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 23739	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 23740	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 23741	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 23742	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 23743	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	mulcncf 23744*	The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋–cn→ℂ))
Theorem	divcncf 23745*	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 23746*	Lemma for pmltpc 23748. (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 23747*	Lemma for pmltpc 23748. (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 23748*	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 23749*	Lemma for ivth 23752. 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 23750*	Lemma for ivth 23752. 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 23751*	Lemma for ivth 23752, 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 23752*	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 23753*	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 23754*	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 23755*	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 23756*	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 23757*	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 23758*	Combine ivthicc 23756 with evthicc 23757 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 23759*	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 23760	Extend class notation with the outer Lebesgue measure.
class vol*
Syntax	cvol 23761	Extend class notation with the Lebesgue measure.
class vol
Definition	df-ovol 23762*	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 23763*	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 23764	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 23765*	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 23766*	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 23767	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 23768	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 23769	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 23770	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 23771*	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 23772	Lemma for elovolm 23773 and related theorems. (Contributed by BJ, 23-Jul-2022.)
⊢ (𝐹 ∈ ((𝐴 ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝐹:ℕ⟶(𝐴 ∩ (ℝ × ℝ)))
Theorem	elovolm 23773*	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 23774*	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 23775*	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 23776	The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
Theorem	ovollb 23777	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 23778*	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 23779	The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
Theorem	ovolf 23780	The domain and range 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 23781	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 23782*	Lemma for ovolss 23783. (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 23783	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 23784	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 23785	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 23786*	Lemma for ovollb2 23787. (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 23787	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 23777). (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 23788	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 23789	The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
⊢ (vol*‘ℚ) = 0
Theorem	ovolctb2 23790	The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘𝐴) = 0)
Theorem	ovol0 23791	The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)
⊢ (vol*‘∅) = 0
Theorem	ovolfi 23792	A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ ℝ) → (vol*‘𝐴) = 0)
Theorem	ovolsn 23793	A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)
⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0)
Theorem	ovolunlem1a 23794*	Lemma for ovolun 23797. (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 23795*	Lemma for ovolun 23797. (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 23796	Lemma for ovolun 23797. (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 23797	The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 23803, 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 23798	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 23799*	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 23800*	Lemma for ovoliun 23803. (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))