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Theorem List for Metamath Proof Explorer - 29001-29100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubmatminr1 29001 If we take a submatrix by removing the row 𝐼 and column 𝐽, then the result is the same on the matrix with row 𝐼 and column 𝐽 modified by the minMatR1 operator. (Contributed by Thierry Arnoux, 25-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀𝐵)    &   𝐸 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)       (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝐼(subMat1‘𝐸)𝐽))
 
20.3.9.4  Matrix literals
 
Syntaxclmat 29002 Extend class notation with the literal matrix conversion function.
class litMat
 
Definitiondf-lmat 29003* Define a function converting words of words into matrices. (Contributed by Thierry Arnoux, 28-Aug-2020.)
litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(#‘𝑚)), 𝑗 ∈ (1...(#‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
 
Theoremlmatval 29004* Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.)
(𝑀𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(#‘𝑀)), 𝑗 ∈ (1...(#‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))
 
Theoremlmatfval 29005* Entries of a literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMat‘𝑊)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑊 ∈ Word Word 𝑉)    &   (𝜑 → (#‘𝑊) = 𝑁)    &   ((𝜑𝑖 ∈ (0..^𝑁)) → (#‘(𝑊𝑖)) = 𝑁)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))       (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)))
 
Theoremlmatfvlem 29006* Useful lemma to extract literal matrix entries. Suggested by Mario Carneiro. (Contributed by Thierry Arnoux, 3-Sep-2020.)
𝑀 = (litMat‘𝑊)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑊 ∈ Word Word 𝑉)    &   (𝜑 → (#‘𝑊) = 𝑁)    &   ((𝜑𝑖 ∈ (0..^𝑁)) → (#‘(𝑊𝑖)) = 𝑁)    &   𝐾 ∈ ℕ0    &   𝐿 ∈ ℕ0    &   𝐼𝑁    &   𝐽𝑁    &   (𝐾 + 1) = 𝐼    &   (𝐿 + 1) = 𝐽    &   (𝑊𝐾) = 𝑋    &   (𝜑 → (𝑋𝐿) = 𝑌)       (𝜑 → (𝐼𝑀𝐽) = 𝑌)
 
Theoremlmatcl 29007* Closure of the literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘𝑊)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑊 ∈ Word Word 𝑉)    &   (𝜑 → (#‘𝑊) = 𝑁)    &   ((𝜑𝑖 ∈ (0..^𝑁)) → (#‘(𝑊𝑖)) = 𝑁)    &   𝑉 = (Base‘𝑅)    &   𝑂 = ((1...𝑁) Mat 𝑅)    &   𝑃 = (Base‘𝑂)    &   (𝜑𝑅𝑋)       (𝜑𝑀𝑃)
 
Theoremlmat22lem 29008* Lemma for lmat22e11 29009 and co. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       ((𝜑𝑖 ∈ (0..^2)) → (#‘(⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩‘𝑖)) = 2)
 
Theoremlmat22e11 29009 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (1𝑀1) = 𝐴)
 
Theoremlmat22e12 29010 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (1𝑀2) = 𝐵)
 
Theoremlmat22e21 29011 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (2𝑀1) = 𝐶)
 
Theoremlmat22e22 29012 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (2𝑀2) = 𝐷)
 
Theoremlmat22det 29013 The determinant of a literal 2x2 complex matrix. (Contributed by Thierry Arnoux, 1-Sep-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)    &    · = (.r𝑅)    &    = (-g𝑅)    &   𝑉 = (Base‘𝑅)    &   𝐽 = ((1...2) maDet 𝑅)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (𝐽𝑀) = ((𝐴 · 𝐷) (𝐶 · 𝐵)))
 
20.3.9.5  Laplace expansion of determinants
 
Theoremmdetpmtr1 29014* The determinant of a matrix with permuted rows is the determinant of the original matrix multiplied by the sign of the permutation. (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)𝑀𝑗))       (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀𝐵𝑃𝐺)) → (𝐷𝑀) = (((𝑍𝑆)‘𝑃) · (𝐷𝐸)))
 
Theoremmdetpmtr2 29015* The determinant of a matrix with permuted columns is the determinant of the original matrix multiplied by the sign of the permutation. (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 ↦ (𝑖𝑀(𝑃𝑗)))       (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀𝐵𝑃𝐺)) → (𝐷𝑀) = (((𝑍𝑆)‘𝑃) · (𝐷𝐸)))
 
Theoremmdetpmtr12 29016* 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 29017* 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)))))
 
Theoremmadjusmdetlem1 29018* Lemma for madjusmdet 29022. (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)    &    · = (.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))))
 
Theoremmadjusmdetlem2 29019* Lemma for madjusmdet 29022. (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)    &    · = (.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))
 
Theoremmadjusmdetlem3 29020* Lemma for madjusmdet 29022. (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)    &    · = (.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))
 
Theoremmadjusmdetlem4 29021* Lemma for madjusmdet 29022. (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)    &    · = (.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 29022 Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrixes. (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 29023* 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)))))))
 
20.3.10  Topology
 
20.3.10.1  Open maps
 
Theoremfvproj 29024* Value of a function on pairs, given two projections 𝐹 and 𝐺. (Contributed by Thierry Arnoux, 30-Dec-2019.)
𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
 
Theoremfimaproj 29025* Image of a cartesian product for a function on pairs, given two projections 𝐹 and 𝐺. (Contributed by Thierry Arnoux, 30-Dec-2019.)
𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹𝑋) × (𝐺𝑌)))
 
Theoremtxomap 29026* 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))
 
20.3.10.2  Topology of the unit circle
 
Theoremqtopt1 29027* 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 29028* 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 29029* 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 29030* 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))
 
20.3.10.3  Refinements
 
Theoremreff 29031* 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 defintion of refinement. Note that this definition uses the axiom of choice through ac6sg 9068. (Contributed by Thierry Arnoux, 12-Jan-2020.)
(𝐴𝑉 → (𝐴Ref𝐵 ↔ ( 𝐵 𝐴 ∧ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)))))
 
Theoremlocfinreflem 29032* 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 29033* 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)))
 
20.3.10.4  Open cover refinement property
 
Syntaxccref 29034 The "every open cover has an 𝐴 refinement" predicate.
class CovHasRef𝐴
 
Definitiondf-cref 29035* 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 29036* 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 29037 Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)
 
Theoremcreftop 29038 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 29039* 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 29040* A formulation of crefi 29039 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = 𝐽    &   𝐵 = CovHasRef𝐴    &   (𝑧𝐴𝜑)       ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑𝑧Ref𝐶))
 
Theoremcrefss 29041 The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)
 
Theoremcmpcref 29042 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 29043* 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))
 
20.3.10.5  Lindelöf spaces
 
Syntaxcldlf 29044 Extend class notation with the class of all Lindelöf spaces.
class Ldlf
 
Definitiondf-ldlf 29045 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 29046* 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))
 
20.3.10.6  Paracompact spaces
 
Syntaxcpcmp 29047 Extend class notation with the class of all paracompact topologies.
class Paracomp
 
Definitiondf-pcmp 29048 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 29049 The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
 
Theoremcmppcmp 29050 Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Comp → 𝐽 ∈ Paracomp)
 
Theoremdispcmp 29051 Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝑋𝑉 → 𝒫 𝑋 ∈ Paracomp)
 
Theorempcmplfin 29052* 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 29053* 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)))
 
20.3.10.7  Pseudometrics
 
Syntaxcmetid 29054 Extend class notation with the class of metric identifications.
class ~Met
 
Syntaxcpstm 29055 Extend class notation with the metric induced by a pseudometric.
class pstoMet
 
Definitiondf-metid 29056* 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 29057* 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 29058* Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)})
 
Theoremmetidss 29059 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 29060 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 29061 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 29062 The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)
 
Theorempstmval 29063* 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 29064 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 29065 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 / )))
 
20.3.10.8  Continuity - misc additions
 
Theoremhauseqcn 29066 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)
 
20.3.10.9  Topology of the closed unit
 
Theoremunitsscn 29067 The closed unit is a subset of the set of the complex numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.)
(0[,]1) ⊆ ℂ
 
Theoremelunitrn 29068 The closed unit is a subset of the set of the real numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 21-Dec-2016.)
(𝐴 ∈ (0[,]1) → 𝐴 ∈ ℝ)
 
Theoremelunitcn 29069 The closed unit is a subset of the set of the complext numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 21-Dec-2016.)
(𝐴 ∈ (0[,]1) → 𝐴 ∈ ℂ)
 
Theoremelunitge0 29070 An element of the closed unit is positive Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 20-Dec-2016.)
(𝐴 ∈ (0[,]1) → 0 ≤ 𝐴)
 
Theoremunitssxrge0 29071 The closed unit is a subset of the set of the extended nonnegative reals. Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.)
(0[,]1) ⊆ (0[,]+∞)
 
Theoremunitdivcld 29072 Necessary conditions for a quotient to be in the closed unit. (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 29073 The closed unit forms a topological monoid. (Contributed by Thierry Arnoux, 25-Mar-2017.)
𝐼 = ((mulGrp‘ℂfld) ↾s (0[,]1))       𝐼 ∈ TopMnd
 
20.3.10.10  Topology of ` ( RR X. RR ) `
 
Theoremunicls 29074 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 29075 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 29076 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 29077 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 29078 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 29079 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 29080 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 29081* Lemma for cnre2csqima 29082. (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 29082* 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 29083* 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))
 
20.3.10.11  Order topology - misc. additions
 
Theoremcnvordtrestixx 29084* 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 29085 Domain of the relation of a preset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       (𝐾 ∈ Preset → dom = 𝐵)
 
Theoremprsrn 29086 Range of the relation of a preset. (Contributed by Thierry Arnoux, 11-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       (𝐾 ∈ Preset → ran = 𝐵)
 
Theoremprsss 29087 Relation of a subpreset. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
 
Theoremprsssdm 29088 Domain of a subpreset relation. (Contributed by Thierry Arnoux, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)
 
Theoremordtprsval 29089* Value of the order topology for a preset. (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 ∈ Preset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸𝐹)))))
 
Theoremordtprsuni 29090* 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 ∈ Preset → 𝐵 = ({𝐵} ∪ (𝐸𝐹)))
 
TheoremordtcnvNEW 29091 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 ∈ Preset → (ordTop‘ ) = (ordTop‘ ))
 
TheoremordtrestNEW 29092 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 ∈ Preset ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
 
Theoremordtrest2NEWlem 29093* Lemma for ordtrest2NEW 29094. (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 29094* 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))
 
Theoremordtconlem1 29095* Connectedness in the order topology of a toset. This is the "easy" direction of ordtcon 29096. See also reconnlem1 22344. (Contributed by Thierry Arnoux, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   𝐽 = (ordTop‘ )       ((𝐾 ∈ Toset ∧ 𝐴𝐵) → ((𝐽t 𝐴) ∈ Con → ∀𝑥𝐴𝑦𝐴𝑟𝐵 ((𝑥 𝑟𝑟 𝑦) → 𝑟𝐴)))
 
Theoremordtcon 29096 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‘ )       
 
20.3.10.12  Continuity in topological spaces - misc. additions
 
Theoremmndpluscn 29097* 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 29098 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 29099* 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 29100* 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)
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