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Theorem List for Metamath Proof Explorer - 31101-31200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmdetpmtr12 31101* The determinant of a matrix with permuted rows and columns is the determinant of the original matrix multiplied by the product of the signs of the permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐺 = (Base‘(SymGrp‘𝑁))    &   𝑆 = (pmSgn‘𝑁)    &   𝑍 = (ℤRHom‘𝑅)    &    · = (.r𝑅)    &   𝐸 = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑃𝑖)𝑀(𝑄𝑗)))    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑀𝐵)    &   (𝜑𝑃𝐺)    &   (𝜑𝑄𝐺)       (𝜑 → (𝐷𝑀) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐷𝐸)))

Theoremmdetlap1 31102* 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)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵𝐼𝑁) → (𝐷𝑀) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑀𝑗) · (𝑗(𝐾𝑀)𝐼)))))

𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)    &   𝐺 = (Base‘(SymGrp‘(1...𝑁)))    &   𝑆 = (pmSgn‘(1...𝑁))    &   𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)    &   𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)𝑈(𝑄𝑗)))    &   (𝜑𝑃𝐺)    &   (𝜑𝑄𝐺)    &   (𝜑 → (𝑃𝑁) = 𝐼)    &   (𝜑 → (𝑄𝑁) = 𝐽)    &   (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))       (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)    &   𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))       ((𝜑𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃𝑆)‘𝑋))

𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)    &   𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))    &   𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗𝐽, (𝑗 − 1), 𝑗)))    &   𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗𝑁, (𝑗 − 1), 𝑗)))    &   𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))    &   (𝜑𝑈𝐵)       (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))

𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)    &   𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))    &   𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗𝐽, (𝑗 − 1), 𝑗)))    &   𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗𝑁, (𝑗 − 1), 𝑗)))       (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

Theoremmadjusmdet 31107 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)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)       (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

Theoremmdetlap 31108* 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)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)       (𝜑 → (𝐷𝑀) = (𝑅 Σg (𝑗 ∈ (1...𝑁) ↦ ((𝑍‘(-1↑(𝐼 + 𝑗))) · ((𝐼𝑀𝑗) · (𝐸‘(𝐼(subMat1‘𝑀)𝑗)))))))

20.3.12  Topology

20.3.12.1  Open maps

Theoremtxomap 31109* 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.)
(𝜑𝐹:𝑋𝑍)    &   (𝜑𝐺:𝑌𝑇)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝑀 ∈ (TopOn‘𝑇))    &   ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐿)    &   ((𝜑𝑦𝐾) → (𝐺𝑦) ∈ 𝑀)    &   (𝜑𝐴 ∈ (𝐽 ×t 𝐾))    &   𝐻 = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)       (𝜑 → (𝐻𝐴) ∈ (𝐿 ×t 𝑀))

20.3.12.2  Topology of the unit circle

Theoremqtopt1 31110* If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.)
𝑋 = 𝐽    &   (𝜑𝐽 ∈ Fre)    &   (𝜑𝐹:𝑋onto𝑌)    &   ((𝜑𝑥𝑌) → (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))       (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)

Theoremqtophaus 31111* 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    &    = (𝐹𝐹)    &   𝐻 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)    &   (𝜑𝐽 ∈ Haus)    &   (𝜑𝐹:𝑋onto𝑌)    &   ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ (𝐽 qTop 𝐹))    &   (𝜑 ∈ (Clsd‘(𝐽 ×t 𝐽)))       (𝜑 → (𝐽 qTop 𝐹) ∈ Haus)

Theoremcirctopn 31112* 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‘(𝐹sfld))

Theoremcirccn 31113* 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))

20.3.12.3  Refinements

Theoremreff 31114* 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 9887. (Contributed by Thierry Arnoux, 12-Jan-2020.)
(𝐴𝑉 → (𝐴Ref𝐵 ↔ ( 𝐵 𝐴 ∧ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)))))

Theoremlocfinreflem 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. 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    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑉𝐽)    &   (𝜑𝑉Ref𝑈)    &   (𝜑𝑉 ∈ (LocFin‘𝐽))       (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))

Theoremlocfinref 31116* 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    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑉𝐽)    &   (𝜑𝑉Ref𝑈)    &   (𝜑𝑉 ∈ (LocFin‘𝐽))       (𝜑 → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))

20.3.12.4  Open cover refinement property

Syntaxccref 31117 The "every open cover has an 𝐴 refinement" predicate.
class CovHasRef𝐴

Definitiondf-cref 31118* 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 𝑗𝐴)𝑧Ref𝑦)}

Theoremiscref 31119* 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 𝐽𝐴)𝑧Ref𝑦)))

Theoremcrefeq 31120 Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)

Theoremcreftop 31121 A space where every open cover has an 𝐴 refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ CovHasRef𝐴𝐽 ∈ Top)

Theoremcrefi 31122* 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 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)

Theoremcrefdf 31123* A formulation of crefi 31122 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = 𝐽    &   𝐵 = CovHasRef𝐴    &   (𝑧𝐴𝜑)       ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑𝑧Ref𝐶))

Theoremcrefss 31124 The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)

Theoremcmpcref 31125 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

Theoremcmpfiref 31126* Every open cover of a Compact space has a finite refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
𝑋 = 𝐽       ((𝐽 ∈ Comp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ Fin ∧ 𝑣Ref𝑈))

20.3.12.5  Lindelöf spaces

Syntaxcldlf 31127 Extend class notation with the class of all Lindelöf spaces.
class Ldlf

Definitiondf-ldlf 31128 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{𝑥𝑥 ≼ ω}

Theoremldlfcntref 31129* Every open cover of a Lindelöf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
𝑋 = 𝐽       ((𝐽 ∈ Ldlf ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))

20.3.12.6  Paracompact spaces

Syntaxcpcmp 31130 Extend class notation with the class of all paracompact topologies.
class Paracomp

Definitiondf-pcmp 31131 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 = {𝑗𝑗 ∈ CovHasRef(LocFin‘𝑗)}

Theoremispcmp 31132 The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))

Theoremcmppcmp 31133 Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Comp → 𝐽 ∈ Paracomp)

Theoremdispcmp 31134 Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝑋𝑉 → 𝒫 𝑋 ∈ Paracomp)

Theorempcmplfin 31135* 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 ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))

Theorempcmplfinf 31136* 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 ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))

20.3.12.7  Pseudometrics

Syntaxcmetid 31137 Extend class notation with the class of metric identifications.
class ~Met

Syntaxcpstm 31138 Extend class notation with the metric induced by a pseudometric.
class pstoMet

Definitiondf-metid 31139* Define the metric identification relation for a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
~Met = (𝑑 ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})

Definitiondf-pstm 31140* Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
pstoMet = (𝑑 ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met𝑑)) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}))

Theoremmetidval 31141* Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)})

Theoremmetidss 31142 As a relation, the metric identification is a subset of a Cartesian product. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))

Theoremmetidv 31143 𝐴 and 𝐵 identify by the metric 𝐷 if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))

Theoremmetideq 31144 Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹))

Theoremmetider 31145 The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)

Theorempstmval 31146* Value of the metric induced by a pseudometric 𝐷. (Contributed by Thierry Arnoux, 7-Feb-2018.)
= (~Met𝐷)       (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}))

Theorempstmfval 31147 Function value of the metric induced by a pseudometric 𝐷 (Contributed by Thierry Arnoux, 11-Feb-2018.)
= (~Met𝐷)       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (pstoMet‘𝐷)[𝐵] ) = (𝐴𝐷𝐵))

Theorempstmxmet 31148 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.)
= (~Met𝐷)       (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / )))

Theoremhauseqcn 31149 In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.)
𝑋 = 𝐽    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   (𝜑 → (𝐹𝐴) = (𝐺𝐴))    &   (𝜑𝐴𝑋)    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)       (𝜑𝐹 = 𝐺)

20.3.12.9  Topology of the closed unit interval

Theoremelunitge0 31150 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)

Theoremunitssxrge0 31151 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[,]+∞)

Theoremunitdivcld 31152 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) → (𝐴𝐵 ↔ (𝐴 / 𝐵) ∈ (0[,]1)))

Theoremiistmd 31153 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

20.3.12.10  Topology of ` ( RR X. RR ) `

Theoremunicls 31154 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

Theoremtpr2tp 31155 The usual topology on (ℝ × ℝ) is the product topology of the usual topology on . (Contributed by Thierry Arnoux, 21-Sep-2017.)
𝐽 = (topGen‘ran (,))       (𝐽 ×t 𝐽) ∈ (TopOn‘(ℝ × ℝ))

Theoremtpr2uni 31156 The usual topology on (ℝ × ℝ) is the product topology of the usual topology on . (Contributed by Thierry Arnoux, 21-Sep-2017.)
𝐽 = (topGen‘ran (,))        (𝐽 ×t 𝐽) = (ℝ × ℝ)

Theoremxpinpreima 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 × 𝐵) = (((1st ↾ (V × V)) “ 𝐴) ∩ ((2nd ↾ (V × V)) “ 𝐵))

Theoremxpinpreima2 31158 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.)
((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = (((1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵)))

Theoremsqsscirc1 31159 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))

Theoremsqsscirc2 31160 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‘(𝐵𝐴))) < (𝐷 / 2) ∧ (abs‘(ℑ‘(𝐵𝐴))) < (𝐷 / 2)) → (abs‘(𝐵𝐴)) < 𝐷))

Theoremcnre2csqlem 31161* Lemma for cnre2csqima 31162. (Contributed by Thierry Arnoux, 27-Sep-2017.)
(𝐺 ↾ (ℝ × ℝ)) = (𝐻𝐹)    &   𝐹 Fn (ℝ × ℝ)    &   𝐺 Fn V    &   (𝑥 ∈ (ℝ × ℝ) → (𝐺𝑥) ∈ ℝ)    &   ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹) → (𝐻‘(𝑥𝑦)) = ((𝐻𝑥) − (𝐻𝑦)))       ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((𝐺 ↾ (ℝ × ℝ)) “ (((𝐺𝑋) − 𝐷)(,)((𝐺𝑋) + 𝐷))) → (abs‘(𝐻‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷))

Theoremcnre2csqima 31162* Image of a centered square by the canonical bijection from (ℝ × ℝ) to . (Contributed by Thierry Arnoux, 27-Sep-2017.)
𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))       ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷)) × (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷))) → ((abs‘(ℜ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷)))

Theoremtpr2rico 31163* 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) ∧ 𝑋𝐴) → ∃𝑟𝐵 (𝑋𝑟𝑟𝐴))

20.3.12.11  Order topology - misc. additions

Theoremcnvordtrestixx 31164* 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*    &   ((𝑥𝐴𝑦𝐴) → (𝑥[,]𝑦) ⊆ 𝐴)       ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))

Theoremprsdm 31165 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)

Theoremprsrn 31166 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)

Theoremprsss 31167 Relation of a subproset. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))

Theoremprsssdm 31168 Domain of a subproset relation. (Contributed by Thierry Arnoux, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)

Theoremordtprsval 31169* Value of the order topology for a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   𝐸 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})    &   𝐹 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})       (𝐾 ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸𝐹)))))

Theoremordtprsuni 31170* Value of the order topology. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   𝐸 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})    &   𝐹 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})       (𝐾 ∈ Proset → 𝐵 = ({𝐵} ∪ (𝐸𝐹)))

TheoremordtcnvNEW 31171 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‘ ))

TheoremordtrestNEW 31172 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 ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))

Theoremordtrest2NEWlem 31173* Lemma for ordtrest2NEW 31174. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)} ⊆ 𝐴)       (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))

Theoremordtrest2NEW 31174* 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 , but in other sets like 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))    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)} ⊆ 𝐴)       (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = ((ordTop‘ ) ↾t 𝐴))

Theoremordtconnlem1 31175* Connectedness in the order topology of a toset. This is the "easy" direction of ordtconn 31176. See also reconnlem1 23410. (Contributed by Thierry Arnoux, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   𝐽 = (ordTop‘ )       ((𝐾 ∈ Toset ∧ 𝐴𝐵) → ((𝐽t 𝐴) ∈ Conn → ∀𝑥𝐴𝑦𝐴𝑟𝐵 ((𝑥 𝑟𝑟 𝑦) → 𝑟𝐴)))

Theoremordtconn 31176 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‘ )

20.3.12.12  Continuity in topological spaces - misc. additions

Theoremmndpluscn 31177* 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)    &   ((𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &    + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)        ∈ ((𝐾 ×t 𝐾) Cn 𝐾)

Theoremmhmhmeotmd 31178 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

Theoremrmulccn 31179* Multiplication by a real constant is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.)
𝐽 = (topGen‘ran (,))    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶)) ∈ (𝐽 Cn 𝐽))

Theoremraddcn 31180* 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)

Theoremxrmulc1cn 31181* 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))    &   (𝜑𝐶 ∈ ℝ+)       (𝜑𝐹 ∈ (𝐽 Cn 𝐽))

Theoremfmcncfil 31182 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))

20.3.12.13  Topology of the extended nonnegative real numbers ordered monoid

Theoremxrge0hmph 31183 The extended nonnegative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.)
II ≃ ((ordTop‘ ≤ ) ↾t (0[,]+∞))

Theoremxrge0iifcnv 31184* 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))))

Theoremxrge0iifcv 31185* 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) → (𝐹𝑋) = -(log‘𝑋))

Theoremxrge0iifiso 31186* 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[,]+∞))

Theoremxrge0iifhmeo 31187* 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)

Theoremxrge0iifhom 31188* 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)) = ((𝐹𝑋) +𝑒 (𝐹𝑌)))

Theoremxrge0iif1 31189* 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

Theoremxrge0iifmhm 31190* 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*𝑠s (0[,]+∞)))

Theoremxrge0pluscn 31191* 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)

Theoremxrge0mulc1cn 31192* 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))    &   (𝜑𝐶 ∈ (0[,)+∞))       (𝜑𝐹 ∈ (𝐽 Cn 𝐽))

Theoremxrge0tps 31193 The extended nonnegative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.)
(ℝ*𝑠s (0[,]+∞)) ∈ TopSp

Theoremxrge0topn 31194 The topology of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 20-Jun-2017.)
(TopOpen‘(ℝ*𝑠s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞))

Theoremxrge0haus 31195 The topology of the extended nonnegative real numbers is Hausdorff. (Contributed by Thierry Arnoux, 26-Jul-2017.)
(TopOpen‘(ℝ*𝑠s (0[,]+∞))) ∈ Haus

Theoremxrge0tmd 31196 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*𝑠s (0[,]+∞)) ∈ TopMnd

Theoremxrge0tmdALT 31197 Alternate proof of xrge0tmd 31196. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(ℝ*𝑠s (0[,]+∞)) ∈ TopMnd