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Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrrxip 24001* The inner product of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉 → (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (ℝfld Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))) = (·𝑖𝐻))

Theoremrrxnm 24002* The norm of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉 → (𝑓𝐵 ↦ (√‘(ℝfld Σg (𝑥𝐼 ↦ ((𝑓𝑥)↑2))))) = (norm‘𝐻))

Theoremrrxcph 24003 Generalized Euclidean real spaces are subcomplex pre-Hilbert spaces. (Contributed by Thierry Arnoux, 23-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉𝐻 ∈ ℂPreHil)

Theoremrrxds 24004* The distance over generalized Euclidean spaces. Compare with df-rrn 35213. (Contributed by Thierry Arnoux, 20-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)       (𝐼𝑉 → (𝑓𝐵, 𝑔𝐵 ↦ (√‘(ℝfld Σg (𝑥𝐼 ↦ (((𝑓𝑥) − (𝑔𝑥))↑2))))) = (dist‘𝐻))

Theoremrrxvsca 24005 The scalar product over generalized Euclidean spaces is the componentwise real number multiplication. (Contributed by Thierry Arnoux, 18-Jan-2023.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)    &    = ( ·𝑠𝐻)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝐼)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑋 ∈ (Base‘𝐻))       (𝜑 → ((𝐴 𝑋)‘𝐽) = (𝐴 · (𝑋𝐽)))

Theoremrrxplusgvscavalb 24006* The result of the addition combined with scalar multiplication in a generalized Euclidean space is defined by its coordinate-wise operations. (Contributed by AV, 21-Jan-2023.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (Base‘𝐻)    &    = ( ·𝑠𝐻)    &   (𝜑𝐼𝑉)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    = (+g𝐻)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝑍 = ((𝐴 𝑋) (𝐶 𝑌)) ↔ ∀𝑖𝐼 (𝑍𝑖) = ((𝐴 · (𝑋𝑖)) + (𝐶 · (𝑌𝑖)))))

Theoremrrxsca 24007 The field of real numbers is the scalar field of the generalized real Euclidean space. (Contributed by AV, 15-Jan-2023.)
𝐻 = (ℝ^‘𝐼)       (𝐼𝑉 → (Scalar‘𝐻) = ℝfld)

Theoremrrx0 24008 The zero ("origin") in a generalized real Euclidean space. (Contributed by AV, 11-Feb-2023.)
𝐻 = (ℝ^‘𝐼)    &    0 = (𝐼 × {0})       (𝐼𝑉 → (0g𝐻) = 0 )

Theoremrrx0el 24009 The zero ("origin") in a generalized real Euclidean space is an element of its base set. (Contributed by AV, 11-Feb-2023.)
0 = (𝐼 × {0})    &   𝑃 = (ℝ ↑m 𝐼)       (𝐼𝑉0𝑃)

Theoremcsbren 24010* Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℝ)       (𝜑 → (Σ𝑘𝐴 (𝐵 · 𝐶)↑2) ≤ (Σ𝑘𝐴 (𝐵↑2) · Σ𝑘𝐴 (𝐶↑2)))

Theoremtrirn 24011* Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℝ)       (𝜑 → (√‘Σ𝑘𝐴 ((𝐵 + 𝐶)↑2)) ≤ ((√‘Σ𝑘𝐴 (𝐵↑2)) + (√‘Σ𝑘𝐴 (𝐶↑2))))

Theoremrrxf 24012* Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   (𝜑𝐹𝑋)       (𝜑𝐹:𝐼⟶ℝ)

Theoremrrxfsupp 24013* Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   (𝜑𝐹𝑋)       (𝜑 → (𝐹 supp 0) ∈ Fin)

Theoremrrxsuppss 24014* Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   (𝜑𝐹𝑋)       (𝜑 → (𝐹 supp 0) ⊆ 𝐼)

Theoremrrxmvallem 24015* Support of the function used for building the distance . (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}       ((𝐼𝑉𝐹𝑋𝐺𝑋) → ((𝑘𝐼 ↦ (((𝐹𝑘) − (𝐺𝑘))↑2)) supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0)))

Theoremrrxmval 24016* The value of the Euclidean metric. Compare with rrnmval 35215. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))       ((𝐼𝑉𝐹𝑋𝐺𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹𝑘) − (𝐺𝑘))↑2)))

Theoremrrxmfval 24017* The value of the Euclidean metric. Compare with rrnval 35214. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))       (𝐼𝑉𝐷 = (𝑓𝑋, 𝑔𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓𝑘) − (𝑔𝑘))↑2))))

Theoremrrxmetlem 24018* Lemma for rrxmet 24019. (Contributed by Thierry Arnoux, 5-Jul-2019.)
𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))    &   (𝜑𝐼𝑉)    &   (𝜑𝐹𝑋)    &   (𝜑𝐺𝑋)    &   (𝜑𝐴𝐼)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐴)       (𝜑 → Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹𝑘) − (𝐺𝑘))↑2) = Σ𝑘𝐴 (((𝐹𝑘) − (𝐺𝑘))↑2))

Theoremrrxmet 24019* 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 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))       (𝐼𝑉𝐷 ∈ (Met‘𝑋))

Theoremrrxdstprj1 24020* 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 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}    &   𝐷 = (dist‘(ℝ^‘𝐼))    &   𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))       (((𝐼𝑉𝐴𝐼) ∧ (𝐹𝑋𝐺𝑋)) → ((𝐹𝐴)𝑀(𝐺𝐴)) ≤ (𝐹𝐷𝐺))

Theoremrrxbasefi 24021 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.)
(𝜑𝑋 ∈ Fin)    &   𝐻 = (ℝ^‘𝑋)    &   𝐵 = (Base‘𝐻)       (𝜑𝐵 = (ℝ ↑m 𝑋))

Theoremrrxdsfi 24022* 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) = (𝑓𝐵, 𝑔𝐵 ↦ (√‘Σ𝑘𝐼 (((𝑓𝑘) − (𝑔𝑘))↑2))))

Theoremrrxmetfi 24023 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)))

Theoremrrxdsfival 24024* 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 ∧ 𝐹𝑋𝐺𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘𝐼 (((𝐹𝑘) − (𝐺𝑘))↑2)))

Theoremehlval 24025 Value of the Euclidean space of dimension 𝑁. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐸 = (𝔼hil𝑁)       (𝑁 ∈ ℕ0𝐸 = (ℝ^‘(1...𝑁)))

Theoremehlbase 24026 The base of the Euclidean space is the set of n-tuples of real numbers. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐸 = (𝔼hil𝑁)       (𝑁 ∈ ℕ0 → (ℝ ↑m (1...𝑁)) = (Base‘𝐸))

Theoremehl0base 24027 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) = {∅}

Theoremehl0 24028 The Euclidean space of dimension 0 consists of the neutral element only. (Contributed by AV, 12-Feb-2023.)
𝐸 = (𝔼hil‘0)    &    0 = (0g𝐸)       (Base‘𝐸) = { 0 }

Theoremehleudis 24029* The Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.)
𝐼 = (1...𝑁)    &   𝐸 = (𝔼hil𝑁)    &   𝑋 = (ℝ ↑m 𝐼)    &   𝐷 = (dist‘𝐸)       (𝑁 ∈ ℕ0𝐷 = (𝑓𝑋, 𝑔𝑋 ↦ (√‘Σ𝑘𝐼 (((𝑓𝑘) − (𝑔𝑘))↑2))))

Theoremehleudisval 24030* 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 = (𝔼hil𝑁)    &   𝑋 = (ℝ ↑m 𝐼)    &   𝐷 = (dist‘𝐸)       ((𝑁 ∈ ℕ0𝐹𝑋𝐺𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘𝐼 (((𝐹𝑘) − (𝐺𝑘))↑2)))

Theoremehl1eudis 24031* 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 = (𝑓𝑋, 𝑔𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1))))

Theoremehl1eudisval 24032 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)       ((𝐹𝑋𝐺𝑋) → (𝐹𝐷𝐺) = (abs‘((𝐹‘1) − (𝐺‘1))))

Theoremehl2eudis 24033* 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 = (𝑓𝑋, 𝑔𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))))

Theoremehl2eudisval 24034 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)       ((𝐹𝑋𝐺𝑋) → (𝐹𝐷𝐺) = (√‘((((𝐹‘1) − (𝐺‘1))↑2) + (((𝐹‘2) − (𝐺‘2))↑2))))

12.5.9  Minimizing Vector Theorem

Theoremminveclem1 24035* Lemma for minvec 24047. 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)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))       (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤𝑅 0 ≤ 𝑤))

Theoremminveclem4c 24036* Lemma for minvec 24047. 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)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )       (𝜑𝑆 ∈ ℝ)

Theoremminveclem2 24037* Lemma for minvec 24047. 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)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑𝐾𝑌)    &   (𝜑𝐿𝑌)    &   (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵))    &   (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵))       (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵))

Theoremminveclem3a 24038* Lemma for minvec 24047. 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)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))       (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))

Theoremminveclem3b 24039* Lemma for minvec 24047. 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)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})       (𝜑𝐹 ∈ (fBas‘𝑌))

Theoremminveclem3 24040* Lemma for minvec 24047. 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)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})       (𝜑 → (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))

Theoremminveclem4a 24041* Lemma for minvec 24047. 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)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   𝑃 = (𝐽 fLim (𝑋filGen𝐹))       (𝜑𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))

Theoremminveclem4b 24042* Lemma for minvec 24047. 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)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   𝑃 = (𝐽 fLim (𝑋filGen𝐹))       (𝜑𝑃𝑋)

Theoremminveclem4 24043* Lemma for minvec 24047. 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)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))    &   𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})    &   𝑃 = (𝐽 fLim (𝑋filGen𝐹))    &   𝑇 = (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2))       (𝜑 → ∃𝑥𝑌𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦)))

Theoremminveclem5 24044* Lemma for minvec 24047. Discharge the assumptions in minveclem4 24043. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))       (𝜑 → ∃𝑥𝑌𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦)))

Theoremminveclem6 24045* Lemma for minvec 24047. 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)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))       ((𝜑𝑥𝑌) → (((𝐴𝐷𝑥)↑2) ≤ ((𝑆↑2) + 0) ↔ ∀𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦))))

Theoremminveclem7 24046* Lemma for minvec 24047. 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)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)    &   𝐽 = (TopOpen‘𝑈)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴 𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))       (𝜑 → ∃!𝑥𝑌𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦)))

Theoremminvec 24047* 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)    &    = (-g𝑈)    &   𝑁 = (norm‘𝑈)    &   (𝜑𝑈 ∈ ℂPreHil)    &   (𝜑𝑌 ∈ (LSubSp‘𝑈))    &   (𝜑 → (𝑈s 𝑌) ∈ CMetSp)    &   (𝜑𝐴𝑋)       (𝜑 → ∃!𝑥𝑌𝑦𝑌 (𝑁‘(𝐴 𝑥)) ≤ (𝑁‘(𝐴 𝑦)))

12.5.10  Projection Theorem

Theorempjthlem1 24048* Lemma for pjth 24050. (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)    &    + = (+g𝑊)    &    = (-g𝑊)    &    , = (·𝑖𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ ℂHil)    &   (𝜑𝑈𝐿)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑈)    &   (𝜑 → ∀𝑥𝑈 (𝑁𝐴) ≤ (𝑁‘(𝐴 𝑥)))    &   𝑇 = ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))       (𝜑 → (𝐴 , 𝐵) = 0)

Theorempjthlem2 24049 Lemma for pjth 24050. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &    , = (·𝑖𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ ℂHil)    &   (𝜑𝑈𝐿)    &   (𝜑𝐴𝑉)    &   𝐽 = (TopOpen‘𝑊)    &    = (LSSum‘𝑊)    &   𝑂 = (ocv‘𝑊)    &   (𝜑𝑈 ∈ (Clsd‘𝐽))       (𝜑𝐴 ∈ (𝑈 (𝑂𝑈)))

Theorempjth 24050 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 ∧ 𝑈𝐿𝑈 ∈ (Clsd‘𝐽)) → (𝑈 (𝑂𝑈)) = 𝑉)

Theorempjth2 24051 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 ∧ 𝑈𝐿𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ dom 𝐾)

Theoremcldcss 24052 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 → (𝑈𝐶 ↔ (𝑈𝐿𝑈 ∈ (Clsd‘𝐽))))

Theoremcldcss2 24053 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)))

Theoremhlhil 24054 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

Theoremaddcncf 24055* The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝑋cn→ℂ))

Theoremsubcncf 24056* The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴𝐵)) ∈ (𝑋cn→ℂ))

Theoremmulcncf 24057* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋cn→ℂ))

Theoremdivcncf 24058* The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→(ℂ ∖ {0})))       (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐵)) ∈ (𝑋cn→ℂ))

13.1.1  Intermediate value theorem

Theorempmltpclem1 24059* Lemma for pmltpc 24061. (Contributed by Mario Carneiro, 1-Jul-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐵 < 𝐶)    &   (𝜑 → (((𝐹𝐴) < (𝐹𝐵) ∧ (𝐹𝐶) < (𝐹𝐵)) ∨ ((𝐹𝐵) < (𝐹𝐴) ∧ (𝐹𝐵) < (𝐹𝐶))))       (𝜑 → ∃𝑎𝑆𝑏𝑆𝑐𝑆 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐)))))

Theorempmltpclem2 24060* Lemma for pmltpc 24061. (Contributed by Mario Carneiro, 1-Jul-2014.)
(𝜑𝐹 ∈ (ℝ ↑pm ℝ))    &   (𝜑𝐴 ⊆ dom 𝐹)    &   (𝜑𝑈𝐴)    &   (𝜑𝑉𝐴)    &   (𝜑𝑊𝐴)    &   (𝜑𝑋𝐴)    &   (𝜑𝑈𝑉)    &   (𝜑𝑊𝑋)    &   (𝜑 → ¬ (𝐹𝑈) ≤ (𝐹𝑉))    &   (𝜑 → ¬ (𝐹𝑋) ≤ (𝐹𝑊))       (𝜑 → ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐)))))

Theorempmltpc 24061* 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) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)) ∨ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)) ∨ ∃𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎 < 𝑏𝑏 < 𝑐 ∧ (((𝐹𝑎) < (𝐹𝑏) ∧ (𝐹𝑐) < (𝐹𝑏)) ∨ ((𝐹𝑏) < (𝐹𝑎) ∧ (𝐹𝑏) < (𝐹𝑐))))))

Theoremivthlem1 24062* Lemma for ivth 24065. The set 𝑆 of all 𝑥 values with (𝐹𝑥) less than 𝑈 is lower bounded by 𝐴 and upper bounded by 𝐵. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑥) ≤ 𝑈}       (𝜑 → (𝐴𝑆 ∧ ∀𝑧𝑆 𝑧𝐵))

Theoremivthlem2 24063* Lemma for ivth 24065. 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑥) ≤ 𝑈}    &   𝐶 = sup(𝑆, ℝ, < )       (𝜑 → ¬ (𝐹𝐶) < 𝑈)

Theoremivthlem3 24064* Lemma for ivth 24065, the intermediate value theorem. Show that (𝐹𝐶) 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑥) ≤ 𝑈}    &   𝐶 = sup(𝑆, ℝ, < )       (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ∧ (𝐹𝐶) = 𝑈))

Theoremivth 24065* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))       (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹𝑐) = 𝑈)

Theoremivth2 24066* The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐵) < 𝑈𝑈 < (𝐹𝐴)))       (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹𝑐) = 𝑈)

Theoremivthle 24067* The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) ≤ 𝑈𝑈 ≤ (𝐹𝐵)))       (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹𝑐) = 𝑈)

Theoremivthle2 24068* The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐵) ≤ 𝑈𝑈 ≤ (𝐹𝐴)))       (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)(𝐹𝑐) = 𝑈)

Theoremivthicc 24069* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 ∈ (𝐴[,]𝐵))    &   (𝜑𝑁 ∈ (𝐴[,]𝐵))    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)       (𝜑 → ((𝐹𝑀)[,](𝐹𝑁)) ⊆ ran 𝐹)

Theoremevthicc 24070* Specialization of the Extreme Value Theorem to a closed interval of . (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))       (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑥) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹𝑧) ≤ (𝐹𝑤)))

Theoremevthicc2 24071* Combine ivthicc 24069 with evthicc 24070 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))       (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran 𝐹 = (𝑥[,]𝑦))

Theoremcniccbdd 24072* 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‘(𝐹𝑦)) ≤ 𝑥)

13.2  Integrals

13.2.1  Lebesgue measure

Syntaxcovol 24073 Extend class notation with the outer Lebesgue measure.
class vol*

Syntaxcvol 24074 Extend class notation with the Lebesgue measure.
class vol

Definitiondf-ovol 24075* Define the outer Lebesgue measure for subsets of the reals. Here 𝑓 is a function from the positive integers to pairs 𝑎, 𝑏 with 𝑎𝑏, 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 𝑏𝑎. (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*, < ))

Definitiondf-vol 24076* 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*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))})

Theoremovolfcl 24077 Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))

Theoremovolfioo 24078* 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) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑧𝑧 < (2nd ‘(𝐹𝑛)))))

Theoremovolficc 24079* Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))

Theoremovolficcss 24080 Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)
(𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)

Theoremovolfsval 24081 The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)       ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺𝑁) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))

Theoremovolfsf 24082 Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
𝐺 = ((abs ∘ − ) ∘ 𝐹)       (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞))

Theoremovolsf 24083 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[,)+∞))

Theoremovolval 24084* 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*, < ))

Theoremelovolmlem 24085 Lemma for elovolm 24086 and related theorems. (Contributed by BJ, 23-Jul-2022.)
(𝐹 ∈ ((𝐴 ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶(𝐴 ∩ (ℝ × ℝ)))

Theoremelovolm 24086* 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*, < ))}       (𝐵𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))

Theoremelovolmr 24087* 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)

Theoremovolmge0 24088* 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*, < ))}       (𝐵𝑀 → 0 ≤ 𝐵)

Theoremovolcl 24089 The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)

Theoremovollb 24090 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*, < ))

Theoremovolgelb 24091* 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)))

Theoremovolge0 24092 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))

Theoremovolf 24093 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[,]+∞)

Theoremovollecl 24094 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)

Theoremovolsslem 24095* Lemma for ovolss 24096. (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*, < ))}       ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))

Theoremovolss 24096 The volume of a set is monotone with respect to set inclusion. (Contributed by Mario Carneiro, 16-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))

Theoremovolsscl 24097 If a set is contained in another of bounded measure, it too is bounded. (Contributed by Mario Carneiro, 18-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐴) ∈ ℝ)

Theoremovolssnul 24098 A subset of a nullset is null. (Contributed by Mario Carneiro, 19-Mar-2014.)
((𝐴𝐵𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (vol*‘𝐴) = 0)

Theoremovollb2lem 24099* Lemma for ovollb2 24100. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩)    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐴 ran ([,] ∘ 𝐹))    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)       (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))

Theoremovollb2 24100 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 24090). (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45264
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