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Type | Label | Description |
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Statement | ||
Theorem | mdetlap1 31101* | A Laplace expansion of the determinant of a matrix, using the adjunct (cofactor) matrix. (Contributed by Thierry Arnoux, 16-Aug-2020.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ 𝐾 = (𝑁 maAdju 𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁) → (𝐷‘𝑀) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑀𝑗) · (𝑗(𝐾‘𝑀)𝐼))))) | ||
Theorem | madjusmdetlem1 31102* | Lemma for madjusmdet 31106. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝐺 = (Base‘(SymGrp‘(1...𝑁))) & ⊢ 𝑆 = (pmSgn‘(1...𝑁)) & ⊢ 𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) & ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗))) & ⊢ (𝜑 → 𝑃 ∈ 𝐺) & ⊢ (𝜑 → 𝑄 ∈ 𝐺) & ⊢ (𝜑 → (𝑃‘𝑁) = 𝐼) & ⊢ (𝜑 → (𝑄‘𝑁) = 𝐽) & ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁)) ⇒ ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) | ||
Theorem | madjusmdetlem2 31103* | Lemma for madjusmdet 31106. (Contributed by Thierry Arnoux, 26-Aug-2020.) |
⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃 ∘ ◡𝑆)‘𝑋)) | ||
Theorem | madjusmdetlem3 31104* | Lemma for madjusmdet 31106. (Contributed by Thierry Arnoux, 27-Aug-2020.) |
⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) & ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) & ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) & ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ◡𝑆)‘𝑖)𝑈((𝑄 ∘ ◡𝑇)‘𝑗))) & ⊢ (𝜑 → 𝑈 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁)) | ||
Theorem | madjusmdetlem4 31105* | Lemma for madjusmdet 31106. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) & ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) & ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) ⇒ ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) | ||
Theorem | madjusmdet 31106 | Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrices. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) | ||
Theorem | mdetlap 31107* | Laplace expansion of the determinant of a square matrix. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘𝑀) = (𝑅 Σg (𝑗 ∈ (1...𝑁) ↦ ((𝑍‘(-1↑(𝐼 + 𝑗))) · ((𝐼𝑀𝑗) · (𝐸‘(𝐼(subMat1‘𝑀)𝑗))))))) | ||
Theorem | txomap 31108* | Given two open maps 𝐹 and 𝐺, 𝐻 mapping pairs of sets, is also an open map for the product topology. (Contributed by Thierry Arnoux, 29-Dec-2019.) |
⊢ (𝜑 → 𝐹:𝑋⟶𝑍) & ⊢ (𝜑 → 𝐺:𝑌⟶𝑇) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑇)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐿) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 “ 𝑦) ∈ 𝑀) & ⊢ (𝜑 → 𝐴 ∈ (𝐽 ×t 𝐾)) & ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉) ⇒ ⊢ (𝜑 → (𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀)) | ||
Theorem | qtopt1 31109* | If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐽 ∈ Fre) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)) ⇒ ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre) | ||
Theorem | qtophaus 31110* | If an open map's graph in the product space (𝐽 ×t 𝐽) is closed, then its quotient topology is Hausdorff. (Contributed by Thierry Arnoux, 4-Jan-2020.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ ∼ = (◡𝐹 ∘ 𝐹) & ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉) & ⊢ (𝜑 → 𝐽 ∈ Haus) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹)) & ⊢ (𝜑 → ∼ ∈ (Clsd‘(𝐽 ×t 𝐽))) ⇒ ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Haus) | ||
Theorem | circtopn 31111* | The topology of the unit circle is generated by open intervals of the polar coordinate. (Contributed by Thierry Arnoux, 4-Jan-2020.) |
⊢ 𝐼 = (0[,](2 · π)) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) & ⊢ 𝐶 = (◡abs “ {1}) ⇒ ⊢ (𝐽 qTop 𝐹) = (TopOpen‘(𝐹 “s ℝfld)) | ||
Theorem | circcn 31112* | The function gluing the real line into the unit circle is continuous. (Contributed by Thierry Arnoux, 5-Jan-2020.) |
⊢ 𝐼 = (0[,](2 · π)) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) & ⊢ 𝐶 = (◡abs “ {1}) ⇒ ⊢ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) | ||
Theorem | reff 31113* | For any cover refinement, there exists a function associating with each set in the refinement a set in the original cover containing it. This is sometimes used as a definition of refinement. Note that this definition uses the axiom of choice through ac6sg 9896. (Contributed by Thierry Arnoux, 12-Jan-2020.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))))) | ||
Theorem | locfinreflem 31114* | A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. The solution is constructed by building unions, so the same method can be used to prove a similar theorem about closed covers. (Contributed by Thierry Arnoux, 29-Jan-2020.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → 𝑋 = ∪ 𝑈) & ⊢ (𝜑 → 𝑉 ⊆ 𝐽) & ⊢ (𝜑 → 𝑉Ref𝑈) & ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽)) ⇒ ⊢ (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))) | ||
Theorem | locfinref 31115* | A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. (Contributed by Thierry Arnoux, 31-Jan-2020.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → 𝑋 = ∪ 𝑈) & ⊢ (𝜑 → 𝑉 ⊆ 𝐽) & ⊢ (𝜑 → 𝑉Ref𝑈) & ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽)) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))) | ||
Syntax | ccref 31116 | The "every open cover has an 𝐴 refinement" predicate. |
class CovHasRef𝐴 | ||
Definition | df-cref 31117* | Define a statement "every open cover has an 𝐴 refinement" , where 𝐴 is a property for refinements like "finite", "countable", "point finite" or "locally finite". (Contributed by Thierry Arnoux, 7-Jan-2020.) |
⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)} | ||
Theorem | iscref 31118* | The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))) | ||
Theorem | crefeq 31119 | Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
⊢ (𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵) | ||
Theorem | creftop 31120 | A space where every open cover has an 𝐴 refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top) | ||
Theorem | crefi 31121* | The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶) | ||
Theorem | crefdf 31122* | A formulation of crefi 31121 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐵 = CovHasRef𝐴 & ⊢ (𝑧 ∈ 𝐴 → 𝜑) ⇒ ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶)) | ||
Theorem | crefss 31123 | The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
⊢ (𝐴 ⊆ 𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵) | ||
Theorem | cmpcref 31124 | Equivalent definition of compact space in terms of open cover refinements. Compact spaces are topologies with finite open cover refinements. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
⊢ Comp = CovHasRefFin | ||
Theorem | cmpfiref 31125* | Every open cover of a Compact space has a finite refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Comp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ Fin ∧ 𝑣Ref𝑈)) | ||
Syntax | cldlf 31126 | Extend class notation with the class of all Lindelöf spaces. |
class Ldlf | ||
Definition | df-ldlf 31127 | Definition of a Lindelöf space. A Lindelöf space is a topological space in which every open cover has a countable subcover. Definition 1 of [BourbakiTop2] p. 195. (Contributed by Thierry Arnoux, 30-Jan-2020.) |
⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω} | ||
Theorem | ldlfcntref 31128* | Every open cover of a Lindelöf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈)) | ||
Syntax | cpcmp 31129 | Extend class notation with the class of all paracompact topologies. |
class Paracomp | ||
Definition | df-pcmp 31130 | Definition of a paracompact topology. A topology is said to be paracompact iff every open cover has an open refinement that is locally finite. The definition 6 of [BourbakiTop1] p. I.69. also requires the topology to be Hausdorff, but this is dropped here. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)} | ||
Theorem | ispcmp 31131 | The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.) |
⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)) | ||
Theorem | cmppcmp 31132 | Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
⊢ (𝐽 ∈ Comp → 𝐽 ∈ Paracomp) | ||
Theorem | dispcmp 31133 | Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Paracomp) | ||
Theorem | pcmplfin 31134* | Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement 𝑣 that is locally finite. (Contributed by Thierry Arnoux, 31-Jan-2020.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) | ||
Theorem | pcmplfinf 31135* | Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement ran 𝑓 that is locally finite, using the same index as the original cover 𝑈. (Contributed by Thierry Arnoux, 31-Jan-2020.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))) | ||
Syntax | cmetid 31136 | Extend class notation with the class of metric identifications. |
class ~Met | ||
Syntax | cpstm 31137 | Extend class notation with the metric induced by a pseudometric. |
class pstoMet | ||
Definition | df-metid 31138* | Define the metric identification relation for a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ ~Met = (𝑑 ∈ ∪ ran PsMet ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)}) | ||
Definition | df-pstm 31139* | Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)})) | ||
Theorem | metidval 31140* | Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)}) | ||
Theorem | metidss 31141 | As a relation, the metric identification is a subset of a Cartesian product. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (𝑋 × 𝑋)) | ||
Theorem | metidv 31142 | 𝐴 and 𝐵 identify by the metric 𝐷 if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0)) | ||
Theorem | metideq 31143 | Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹)) | ||
Theorem | metider 31144 | The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) Er 𝑋) | ||
Theorem | pstmval 31145* | Value of the metric induced by a pseudometric 𝐷. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
⊢ ∼ = (~Met‘𝐷) ⇒ ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)})) | ||
Theorem | pstmfval 31146 | Function value of the metric induced by a pseudometric 𝐷 (Contributed by Thierry Arnoux, 11-Feb-2018.) |
⊢ ∼ = (~Met‘𝐷) ⇒ ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ([𝐴] ∼ (pstoMet‘𝐷)[𝐵] ∼ ) = (𝐴𝐷𝐵)) | ||
Theorem | pstmxmet 31147 | The metric induced by a pseudometric is a full-fledged metric on the equivalence classes of the metric identification. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
⊢ ∼ = (~Met‘𝐷) ⇒ ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / ∼ ))) | ||
Theorem | hauseqcn 31148 | In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐾 ∈ Haus) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | elunitge0 31149 | An element of the closed unit interval is positive. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 20-Dec-2016.) |
⊢ (𝐴 ∈ (0[,]1) → 0 ≤ 𝐴) | ||
Theorem | unitssxrge0 31150 | The closed unit interval is a subset of the set of the extended nonnegative reals. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 12-Dec-2016.) |
⊢ (0[,]1) ⊆ (0[,]+∞) | ||
Theorem | unitdivcld 31151 | Necessary conditions for a quotient to be in the closed unit interval. (somewhat too strong, it would be sufficient that A and B are in RR+) (Contributed by Thierry Arnoux, 20-Dec-2016.) |
⊢ ((𝐴 ∈ (0[,]1) ∧ 𝐵 ∈ (0[,]1) ∧ 𝐵 ≠ 0) → (𝐴 ≤ 𝐵 ↔ (𝐴 / 𝐵) ∈ (0[,]1))) | ||
Theorem | iistmd 31152 | The closed unit interval forms a topological monoid under multiplication. (Contributed by Thierry Arnoux, 25-Mar-2017.) |
⊢ 𝐼 = ((mulGrp‘ℂfld) ↾s (0[,]1)) ⇒ ⊢ 𝐼 ∈ TopMnd | ||
Theorem | unicls 31153 | The union of the closed set is the underlying set of the topology. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ∪ (Clsd‘𝐽) = 𝑋 | ||
Theorem | tpr2tp 31154 | The usual topology on (ℝ × ℝ) is the product topology of the usual topology on ℝ. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ (𝐽 ×t 𝐽) ∈ (TopOn‘(ℝ × ℝ)) | ||
Theorem | tpr2uni 31155 | The usual topology on (ℝ × ℝ) is the product topology of the usual topology on ℝ. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ ∪ (𝐽 ×t 𝐽) = (ℝ × ℝ) | ||
Theorem | xpinpreima 31156 | Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
⊢ (𝐴 × 𝐵) = ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵)) | ||
Theorem | xpinpreima2 31157 | Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ((◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵))) | ||
Theorem | sqsscirc1 31158 | The complex square of side 𝐷 is a subset of the complex circle of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.) |
⊢ ((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ+) → ((𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2)) → (√‘((𝑋↑2) + (𝑌↑2))) < 𝐷)) | ||
Theorem | sqsscirc2 31159 | The complex square of side 𝐷 is a subset of the complex disc of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐷 ∈ ℝ+) → (((abs‘(ℜ‘(𝐵 − 𝐴))) < (𝐷 / 2) ∧ (abs‘(ℑ‘(𝐵 − 𝐴))) < (𝐷 / 2)) → (abs‘(𝐵 − 𝐴)) < 𝐷)) | ||
Theorem | cnre2csqlem 31160* | Lemma for cnre2csqima 31161. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
⊢ (𝐺 ↾ (ℝ × ℝ)) = (𝐻 ∘ 𝐹) & ⊢ 𝐹 Fn (ℝ × ℝ) & ⊢ 𝐺 Fn V & ⊢ (𝑥 ∈ (ℝ × ℝ) → (𝐺‘𝑥) ∈ ℝ) & ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹) → (𝐻‘(𝑥 − 𝑦)) = ((𝐻‘𝑥) − (𝐻‘𝑦))) ⇒ ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ (◡(𝐺 ↾ (ℝ × ℝ)) “ (((𝐺‘𝑋) − 𝐷)(,)((𝐺‘𝑋) + 𝐷))) → (abs‘(𝐻‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷)) | ||
Theorem | cnre2csqima 31161* | Image of a centered square by the canonical bijection from (ℝ × ℝ) to ℂ. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ⇒ ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) × (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷))) → ((abs‘(ℜ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷))) | ||
Theorem | tpr2rico 31162* | For any point of an open set of the usual topology on (ℝ × ℝ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the (𝑙↑+∞) norm 𝑋. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣))) & ⊢ 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) ⇒ ⊢ ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴)) | ||
Theorem | cnvordtrestixx 31163* | The restriction of the 'greater than' order to an interval gives the same topology as the subspace topology. (Contributed by Thierry Arnoux, 1-Apr-2017.) |
⊢ 𝐴 ⊆ ℝ* & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) ⇒ ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) | ||
Theorem | prsdm 31164 | Domain of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵) | ||
Theorem | prsrn 31165 | Range of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ (𝐾 ∈ Proset → ran ≤ = 𝐵) | ||
Theorem | prsss 31166 | Relation of a subproset. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) | ||
Theorem | prsssdm 31167 | Domain of a subproset relation. (Contributed by Thierry Arnoux, 12-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴) | ||
Theorem | ordtprsval 31168* | Value of the order topology for a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) & ⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}) ⇒ ⊢ (𝐾 ∈ Proset → (ordTop‘ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸 ∪ 𝐹))))) | ||
Theorem | ordtprsuni 31169* | Value of the order topology. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) & ⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}) ⇒ ⊢ (𝐾 ∈ Proset → 𝐵 = ∪ ({𝐵} ∪ (𝐸 ∪ 𝐹))) | ||
Theorem | ordtcnvNEW 31170 | The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.) (Revised by Thierry Arnoux, 13-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ (𝐾 ∈ Proset → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ )) | ||
Theorem | ordtrestNEW 31171 | The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴)) | ||
Theorem | ordtrest2NEWlem 31172* | Lemma for ordtrest2NEW 31173. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ (𝜑 → 𝐾 ∈ Toset) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) ⇒ ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))) | ||
Theorem | ordtrest2NEW 31173* | An interval-closed set 𝐴 in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in ℝ, but in other sets like ℚ there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ (𝜑 → 𝐾 ∈ Toset) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) ⇒ ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾t 𝐴)) | ||
Theorem | ordtconnlem1 31174* | Connectedness in the order topology of a toset. This is the "easy" direction of ordtconn 31175. See also reconnlem1 23417. (Contributed by Thierry Arnoux, 14-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ 𝐽 = (ordTop‘ ≤ ) ⇒ ⊢ ((𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵) → ((𝐽 ↾t 𝐴) ∈ Conn → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑟 ∈ 𝐵 ((𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦) → 𝑟 ∈ 𝐴))) | ||
Theorem | ordtconn 31175 | Connectedness in the order topology of a complete uniform totally ordered space. (Contributed by Thierry Arnoux, 15-Sep-2018.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ 𝐽 = (ordTop‘ ≤ ) ⇒ ⊢ ⊤ | ||
Theorem | mndpluscn 31176* | A mapping that is both a homeomorphism and a monoid homomorphism preserves the "continuousness" of the operation. (Contributed by Thierry Arnoux, 25-Mar-2017.) |
⊢ 𝐹 ∈ (𝐽Homeo𝐾) & ⊢ + :(𝐵 × 𝐵)⟶𝐵 & ⊢ ∗ :(𝐶 × 𝐶)⟶𝐶 & ⊢ 𝐽 ∈ (TopOn‘𝐵) & ⊢ 𝐾 ∈ (TopOn‘𝐶) & ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))) & ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ⇒ ⊢ ∗ ∈ ((𝐾 ×t 𝐾) Cn 𝐾) | ||
Theorem | mhmhmeotmd 31177 | Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.) |
⊢ 𝐹 ∈ (𝑆 MndHom 𝑇) & ⊢ 𝐹 ∈ ((TopOpen‘𝑆)Homeo(TopOpen‘𝑇)) & ⊢ 𝑆 ∈ TopMnd & ⊢ 𝑇 ∈ TopSp ⇒ ⊢ 𝑇 ∈ TopMnd | ||
Theorem | rmulccn 31178* | Multiplication by a real constant is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶)) ∈ (𝐽 Cn 𝐽)) | ||
Theorem | raddcn 31179* | Addition in the real numbers is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) | ||
Theorem | xrmulc1cn 31180* | The operation multiplying an extended real number by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.) |
⊢ 𝐽 = (ordTop‘ ≤ ) & ⊢ 𝐹 = (𝑥 ∈ ℝ* ↦ (𝑥 ·e 𝐶)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐽)) | ||
Theorem | fmcncfil 31181 | The image of a Cauchy filter by a continuous filter map is a Cauchy filter. (Contributed by Thierry Arnoux, 12-Nov-2017.) |
⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = (MetOpen‘𝐸) ⇒ ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐸 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝐵 ∈ (CauFil‘𝐷)) → ((𝑌 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐸)) | ||
Theorem | xrge0hmph 31182 | The extended nonnegative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.) |
⊢ II ≃ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | ||
Theorem | xrge0iifcnv 31183* | Define a bijection from [0, 1] onto [0, +∞]. (Contributed by Thierry Arnoux, 29-Mar-2017.) |
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) ⇒ ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦)))) | ||
Theorem | xrge0iifcv 31184* | The defined function's value in the real. (Contributed by Thierry Arnoux, 1-Apr-2017.) |
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) ⇒ ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋)) | ||
Theorem | xrge0iifiso 31185* | The defined bijection from the closed unit interval onto the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.) |
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) ⇒ ⊢ 𝐹 Isom < , ◡ < ((0[,]1), (0[,]+∞)) | ||
Theorem | xrge0iifhmeo 31186* | Expose a homeomorphism from the closed unit interval to the extended nonnegative reals. (Contributed by Thierry Arnoux, 1-Apr-2017.) |
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) & ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ⇒ ⊢ 𝐹 ∈ (IIHomeo𝐽) | ||
Theorem | xrge0iifhom 31187* | The defined function from the closed unit interval to the extended nonnegative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 5-Apr-2017.) |
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) & ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ⇒ ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌))) | ||
Theorem | xrge0iif1 31188* | Condition for the defined function, -(log‘𝑥) to be a monoid homomorphism. (Contributed by Thierry Arnoux, 20-Jun-2017.) |
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) & ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ⇒ ⊢ (𝐹‘1) = 0 | ||
Theorem | xrge0iifmhm 31189* | The defined function from the closed unit interval to the extended nonnegative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.) |
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) & ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ⇒ ⊢ 𝐹 ∈ (((mulGrp‘ℂfld) ↾s (0[,]1)) MndHom (ℝ*𝑠 ↾s (0[,]+∞))) | ||
Theorem | xrge0pluscn 31190* | The addition operation of the extended nonnegative real numbers monoid is continuous. (Contributed by Thierry Arnoux, 24-Mar-2017.) |
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) & ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) & ⊢ + = ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) ⇒ ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽) | ||
Theorem | xrge0mulc1cn 31191* | The operation multiplying a nonnegative real numbers by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 6-Jul-2017.) |
⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) & ⊢ 𝐹 = (𝑥 ∈ (0[,]+∞) ↦ (𝑥 ·e 𝐶)) & ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐽)) | ||
Theorem | xrge0tps 31192 | The extended nonnegative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.) |
⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | ||
Theorem | xrge0topn 31193 | The topology of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 20-Jun-2017.) |
⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | ||
Theorem | xrge0haus 31194 | The topology of the extended nonnegative real numbers is Hausdorff. (Contributed by Thierry Arnoux, 26-Jul-2017.) |
⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) ∈ Haus | ||
Theorem | xrge0tmd 31195 | The extended nonnegative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof Shortened by Thierry Arnoux, 21-Jun-2017.) |
⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd | ||
Theorem | xrge0tmdALT 31196 | Alternate proof of xrge0tmd 31195. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd | ||
Theorem | lmlim 31197 | Relate a limit in a given topology to a complex number limit, provided that topology agrees with the common topology on ℂ on the required subset. (Contributed by Thierry Arnoux, 11-Jul-2017.) |
⊢ 𝐽 ∈ (TopOn‘𝑌) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ (𝐽 ↾t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋) & ⊢ 𝑋 ⊆ ℂ ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) | ||
Theorem | lmlimxrge0 31198 | Relate a limit in the nonnegative extended reals to a complex limit, provided the considered function is a real function. (Contributed by Thierry Arnoux, 11-Jul-2017.) |
⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ 𝑋 ⊆ (0[,)+∞) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) | ||
Theorem | rge0scvg 31199 | Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 16240. (Contributed by Thierry Arnoux, 28-Jul-2017.) |
⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → sup(ran seq1( + , 𝐹), ℝ, < ) ∈ ℝ) | ||
Theorem | fsumcvg4 31200 | A serie with finite support is a finite sum, and therefore converges. (Contributed by Thierry Arnoux, 6-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
⊢ 𝑆 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑆⟶ℂ) & ⊢ (𝜑 → (◡𝐹 “ (ℂ ∖ {0})) ∈ Fin) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
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