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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rrxsca 24001 | The field of real numbers is the scalar field of the generalized real Euclidean space. (Contributed by AV, 15-Jan-2023.) |
⊢ 𝐻 = (ℝ^‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → (Scalar‘𝐻) = ℝfld) | ||
Theorem | rrx0 24002 | The zero ("origin") in a generalized real Euclidean space. (Contributed by AV, 11-Feb-2023.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 0 = (𝐼 × {0}) ⇒ ⊢ (𝐼 ∈ 𝑉 → (0g‘𝐻) = 0 ) | ||
Theorem | rrx0el 24003 | 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) ⇒ ⊢ (𝐼 ∈ 𝑉 → 0 ∈ 𝑃) | ||
Theorem | csbren 24004* | Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶)↑2) ≤ (Σ𝑘 ∈ 𝐴 (𝐵↑2) · Σ𝑘 ∈ 𝐴 (𝐶↑2))) | ||
Theorem | trirn 24005* | Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (√‘Σ𝑘 ∈ 𝐴 ((𝐵 + 𝐶)↑2)) ≤ ((√‘Σ𝑘 ∈ 𝐴 (𝐵↑2)) + (√‘Σ𝑘 ∈ 𝐴 (𝐶↑2)))) | ||
Theorem | rrxf 24006* | 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 24007* | 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 24008* | 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 24009* | 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 24010* | The value of the Euclidean metric. Compare with rrnmval 35108. (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 24011* | The value of the Euclidean metric. Compare with rrnval 35107. (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 24012* | Lemma for rrxmet 24013. (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 24013* | 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 24014* | 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 24015 | 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 24016* | 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 24017 | 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 24018* | 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 24019 | 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 24020 | 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 24021 | 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 24022 | 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 24023* | 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 24024* | 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 24025* | 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 24026 | 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 24027* | 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 24028 | 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)))) | ||
Theorem | minveclem1 24029* | Lemma for minvec 24041. 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 24030* | Lemma for minvec 24041. 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 24031* | Lemma for minvec 24041. 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 24032* | Lemma for minvec 24041. 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 24033* | Lemma for minvec 24041. 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 24034* | Lemma for minvec 24041. 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 24035* | Lemma for minvec 24041. 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 24036* | Lemma for minvec 24041. 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 24037* | Lemma for minvec 24041. 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 24038* | Lemma for minvec 24041. Discharge the assumptions in minveclem4 24037. (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 24039* | Lemma for minvec 24041. 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 24040* | Lemma for minvec 24041. 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 24041* | 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))) | ||
Theorem | pjthlem1 24042* | Lemma for pjth 24044. (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 24043 | Lemma for pjth 24044. (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 24044 | 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 24045 | 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 24046 | 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 24047 | 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 24048 | 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) | ||
Theorem | mulcncf 24049* | 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 24050* | 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)) | ||
Theorem | pmltpclem1 24051* | Lemma for pmltpc 24053. (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 24052* | Lemma for pmltpc 24053. (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 24053* | 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 24054* | Lemma for ivth 24057. 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 24055* | Lemma for ivth 24057. 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 24056* | Lemma for ivth 24057, 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 24057* | 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 24058* | 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 24059* | 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 24060* | 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 24061* | 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 24062* | 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 24063* | Combine ivthicc 24061 with evthicc 24062 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 24064* | 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) | ||
Syntax | covol 24065 | Extend class notation with the outer Lebesgue measure. |
class vol* | ||
Syntax | cvol 24066 | Extend class notation with the Lebesgue measure. |
class vol | ||
Definition | df-ovol 24067* | 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 24068* | 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 24069 | 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 24070* | 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 24071* | 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 24072 | 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 24073 | 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 24074 | 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 24075 | 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 24076* | 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 24077 | Lemma for elovolm 24078 and related theorems. (Contributed by BJ, 23-Jul-2022.) |
⊢ (𝐹 ∈ ((𝐴 ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶(𝐴 ∩ (ℝ × ℝ))) | ||
Theorem | elovolm 24078* | 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 24079* | 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 24080* | 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 24081 | The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*) | ||
Theorem | ovollb 24082 | 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 24083* | 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 24084 | The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.) |
⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴)) | ||
Theorem | ovolf 24085 | 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 24086 | 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 24087* | Lemma for ovolss 24088. (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 24088 | 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 24089 | 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 24090 | 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 24091* | Lemma for ovollb2 24092. (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 24092 | 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 24082). (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 24093 | 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 24094 | The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.) |
⊢ (vol*‘ℚ) = 0 | ||
Theorem | ovolctb2 24095 | The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘𝐴) = 0) | ||
Theorem | ovol0 24096 | The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.) |
⊢ (vol*‘∅) = 0 | ||
Theorem | ovolfi 24097 | A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ ℝ) → (vol*‘𝐴) = 0) | ||
Theorem | ovolsn 24098 | A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0) | ||
Theorem | ovolunlem1a 24099* | Lemma for ovolun 24102. (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 24100* | Lemma for ovolun 24102. (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)) |
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