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Theorem List for Metamath Proof Explorer - 30401-30500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempsgnfzto1st 30401* The permutation sign for moving one element to the first position. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑆 = (pmSgn‘𝐷)       (𝐼𝐷 → (𝑆𝑃) = (-1↑(𝐼 + 1)))

20.3.10.3  Transpositions

Theorempmtridf1o 30402 Transpositions of 𝑋 and 𝑌 (understood to be the identity when 𝑋 = 𝑌), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑𝑇:𝐴1-1-onto𝐴)

Theorempmtridfv1 30403 Value at X of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑 → (𝑇𝑋) = 𝑌)

Theorempmtridfv2 30404 Value at Y of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑 → (𝑇𝑌) = 𝑋)

20.3.10.4  Submatrices

Syntaxcsmat 30405 Syntax for a function generating submatrices.
class subMat1

Definitiondf-smat 30406* Define a function generating submatrices of an integer-indexed matrix. The function maps an index in ((1...𝑀) × (1...𝑁)) into a new index in ((1...(𝑀 − 1)) × (1...(𝑁 − 1))). A submatrix is obtained by deleting a row and a column of the original matrix. Because this function re-indexes the matrix, the resulting submatrix still has the same index set for rows and columns, and its determinent is defined, unlike the current df-subma 20752. (Contributed by Thierry Arnoux, 18-Aug-2020.)
subMat1 = (𝑚 ∈ V ↦ (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))

Theoremsmatfval 30407* Value of the submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → (𝐾(subMat1‘𝑀)𝐿) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))

Theoremsmatrcl 30408 Closure of the rectangular submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))       (𝜑𝑆 ∈ (𝐵𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))

Theoremsmatlem 30409 Lemma for the next theorems. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐽 ∈ ℕ)    &   (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝑋)    &   (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝑌)       (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))

Theoremsmattl 30410 Entries of a submatrix, top left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (1..^𝐾))    &   (𝜑𝐽 ∈ (1..^𝐿))       (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽))

Theoremsmattr 30411 Entries of a submatrix, top right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (𝐾...𝑀))    &   (𝜑𝐽 ∈ (1..^𝐿))       (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽))

Theoremsmatbl 30412 Entries of a submatrix, bottom left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (1..^𝐾))    &   (𝜑𝐽 ∈ (𝐿...𝑁))       (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴(𝐽 + 1)))

Theoremsmatbr 30413 Entries of a submatrix, bottom right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (𝐾...𝑀))    &   (𝜑𝐽 ∈ (𝐿...𝑁))       (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴(𝐽 + 1)))

Theoremsmatcl 30414 Closure of the square submatrix: if 𝑀 is a square matrix of dimension 𝑁 with indices in (1...𝑁), then a submatrix of 𝑀 is of dimension (𝑁 − 1). (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅))    &   𝑆 = (𝐾(subMat1‘𝑀)𝐿)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑁))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)       (𝜑𝑆𝐶)

Theoremmatmpt2 30415* Write a square matrix as a mapping operation. (Contributed by Thierry Arnoux, 16-Aug-2020.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵𝑀 = (𝑖𝑁, 𝑗𝑁 ↦ (𝑖𝑀𝑗)))

Theorem1smat1 30416 The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 20758. (Contributed by Thierry Arnoux, 19-Aug-2020.)
1 = (1r‘((1...𝑁) Mat 𝑅))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ (1...𝑁))       (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)))

Theoremsubmat1n 30417 One case where the submatrix with integer indices, subMat1, and the general submatrix subMat, agree. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑁 ∈ ℕ ∧ 𝑀𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑀)𝑁))

Theoremsubmatres 30418 Special case where the submatrix is a restriction of the initial matrix, and no renumbering occurs. (Contributed by Thierry Arnoux, 26-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑁 ∈ ℕ ∧ 𝑀𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑀 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))

Theoremsubmateqlem1 30419 Lemma for submateq 30421. (Contributed by Thierry Arnoux, 25-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑁))    &   (𝜑𝑀 ∈ (1...(𝑁 − 1)))    &   (𝜑𝐾𝑀)       (𝜑 → (𝑀 ∈ (𝐾...𝑁) ∧ (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾})))

Theoremsubmateqlem2 30420 Lemma for submateq 30421. (Contributed by Thierry Arnoux, 26-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑁))    &   (𝜑𝑀 ∈ (1...(𝑁 − 1)))    &   (𝜑𝑀 < 𝐾)       (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾})))

Theoremsubmateq 30421* Sufficient condition for two submatrices to be equal. (Contributed by Thierry Arnoux, 25-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)    &   ((𝜑𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗))       (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽))

Theoremsubmatminr1 30422 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))

20.3.10.5  Matrix literals

Syntaxclmat 30423 Extend class notation with the literal matrix conversion function.
class litMat

Definitiondf-lmat 30424* 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 30425* 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 30426* 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 30427* 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 30428* 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 30429* Lemma for lmat22e11 30430 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 30430 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 30431 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 30432 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 30433 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 30434 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)))

20.3.10.6  Laplace expansion of determinants

Theoremmdetpmtr1 30435* 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 30436* 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 30437* 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 30438* 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 30443 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 30444* 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)))))))

20.3.11  Topology

20.3.11.1  Open maps

Theoremfvproj 30445* 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 30446* 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 30447* 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))

20.3.11.2  Topology of the unit circle

Theoremqtopt1 30448* 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 30449* 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 30450* 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 30451* 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))

20.3.11.3  Refinements

Theoremreff 30452* 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 9626. (Contributed by Thierry Arnoux, 12-Jan-2020.)
(𝐴𝑉 → (𝐴Ref𝐵 ↔ ( 𝐵 𝐴 ∧ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)))))

Theoremlocfinreflem 30453* 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 30454* 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)))

20.3.11.4  Open cover refinement property

Syntaxccref 30455 The "every open cover has an 𝐴 refinement" predicate.
class CovHasRef𝐴

Definitiondf-cref 30456* 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 30457* 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 30458 Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)

Theoremcreftop 30459 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 30460* 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 30461* A formulation of crefi 30460 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = 𝐽    &   𝐵 = CovHasRef𝐴    &   (𝑧𝐴𝜑)       ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑𝑧Ref𝐶))

Theoremcrefss 30462 The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)

Theoremcmpcref 30463 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 30464* 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))

20.3.11.5  Lindelöf spaces

Syntaxcldlf 30465 Extend class notation with the class of all Lindelöf spaces.
class Ldlf

Definitiondf-ldlf 30466 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 30467* 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))

20.3.11.6  Paracompact spaces

Syntaxcpcmp 30468 Extend class notation with the class of all paracompact topologies.
class Paracomp

Definitiondf-pcmp 30469 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 30470 The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))

Theoremcmppcmp 30471 Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Comp → 𝐽 ∈ Paracomp)

Theoremdispcmp 30472 Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝑋𝑉 → 𝒫 𝑋 ∈ Paracomp)

Theorempcmplfin 30473* 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 30474* 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)))

20.3.11.7  Pseudometrics

Syntaxcmetid 30475 Extend class notation with the class of metric identifications.
class ~Met

Syntaxcpstm 30476 Extend class notation with the metric induced by a pseudometric.
class pstoMet

Definitiondf-metid 30477* 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 30478* 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 30479* Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)})

Theoremmetidss 30480 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 30481 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 30482 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 30483 The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)

Theorempstmval 30484* 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 30485 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 30486 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 30487 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)

20.3.11.9  Topology of the closed unit interval

Theoremunitsscn 30488 The closed unit interval is a subset of the set of the complex numbers. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 12-Dec-2016.)
(0[,]1) ⊆ ℂ

Theoremelunitrn 30489 The closed unit interval is a subset of the set of the real numbers. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 21-Dec-2016.)
(𝐴 ∈ (0[,]1) → 𝐴 ∈ ℝ)

Theoremelunitcn 30490 The closed unit interval is a subset of the set of the complext numbers. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 21-Dec-2016.)
(𝐴 ∈ (0[,]1) → 𝐴 ∈ ℂ)

Theoremelunitge0 30491 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 30492 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 30493 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 30494 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

20.3.11.10  Topology of ` ( RR X. RR ) `

Theoremunicls 30495 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 30496 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 30497 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 30498 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 30499 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 30500 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))

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