HomeHome Metamath Proof Explorer
Theorem List (p. 322 of 470)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-29646)
  Hilbert Space Explorer  Hilbert Space Explorer
(29647-31169)
  Users' Mathboxes  Users' Mathboxes
(31170-46948)
 

Theorem List for Metamath Proof Explorer - 32101-32200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcextdg 32101 Syntax for the field extension degree operation.
class [:]
 
Definitiondf-fldext 32102* Definition of the field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
/FldExt = {βŸ¨π‘’, π‘“βŸ© ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 β†Ύs (Baseβ€˜π‘“)) ∧ (Baseβ€˜π‘“) ∈ (SubRingβ€˜π‘’)))}
 
Definitiondf-extdg 32103* Definition of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
[:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt β€œ {𝑒}) ↦ (dimβ€˜((subringAlg β€˜π‘’)β€˜(Baseβ€˜π‘“))))
 
Definitiondf-finext 32104* Definition of the finite field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
/FinExt = {βŸ¨π‘’, π‘“βŸ© ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ β„•0)}
 
Definitiondf-algext 32105* Definition of the algebraic extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
/AlgExt = {βŸ¨π‘’, π‘“βŸ© ∣ (𝑒/FldExt𝑓 ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘’)βˆƒπ‘ ∈ (Poly1β€˜π‘“)(((eval1β€˜π‘“)β€˜π‘)β€˜π‘₯) = (0gβ€˜π‘’))}
 
Theoremrelfldext 32106 The field extension is a relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
Rel /FldExt
 
Theorembrfldext 32107 The field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)
((𝐸 ∈ Field ∧ 𝐹 ∈ Field) β†’ (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 β†Ύs (Baseβ€˜πΉ)) ∧ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ))))
 
Theoremccfldextrr 32108 The field of the complex numbers is an extension of the field of the real numbers. (Contributed by Thierry Arnoux, 20-Jul-2023.)
β„‚fld/FldExtℝfld
 
Theoremfldextfld1 32109 A field extension is only defined if the extension is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.)
(𝐸/FldExt𝐹 β†’ 𝐸 ∈ Field)
 
Theoremfldextfld2 32110 A field extension is only defined if the subfield is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.)
(𝐸/FldExt𝐹 β†’ 𝐹 ∈ Field)
 
Theoremfldextsubrg 32111 Field extension implies a subring relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
π‘ˆ = (Baseβ€˜πΉ)    β‡’   (𝐸/FldExt𝐹 β†’ π‘ˆ ∈ (SubRingβ€˜πΈ))
 
Theoremfldextress 32112 Field extension implies a structure restriction relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
(𝐸/FldExt𝐹 β†’ 𝐹 = (𝐸 β†Ύs (Baseβ€˜πΉ)))
 
Theorembrfinext 32113 The finite field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)
(𝐸/FldExt𝐹 β†’ (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ β„•0))
 
Theoremextdgval 32114 Value of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
(𝐸/FldExt𝐹 β†’ (𝐸[:]𝐹) = (dimβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
 
Theoremfldextsralvec 32115 The subring algebra associated with a field extension is a vector space. (Contributed by Thierry Arnoux, 3-Aug-2023.)
(𝐸/FldExt𝐹 β†’ ((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) ∈ LVec)
 
Theoremextdgcl 32116 Closure of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
(𝐸/FldExt𝐹 β†’ (𝐸[:]𝐹) ∈ β„•0*)
 
Theoremextdggt0 32117 Degrees of field extension are greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.)
(𝐸/FldExt𝐹 β†’ 0 < (𝐸[:]𝐹))
 
Theoremfldexttr 32118 Field extension is a transitive relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) β†’ 𝐸/FldExt𝐾)
 
Theoremfldextid 32119 The field extension relation is reflexive. (Contributed by Thierry Arnoux, 30-Jul-2023.)
(𝐹 ∈ Field β†’ 𝐹/FldExt𝐹)
 
Theoremextdgid 32120 A trivial field extension has degree one. (Contributed by Thierry Arnoux, 4-Aug-2023.)
(𝐸 ∈ Field β†’ (𝐸[:]𝐸) = 1)
 
Theoremextdgmul 32121 The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, the degree of the extension 𝐸/FldExt𝐾 is the product of the degrees of the extensions 𝐸/FldExt𝐹 and 𝐹/FldExt𝐾. Proposition 1.2 of [Lang], p. 224. (Contributed by Thierry Arnoux, 30-Jul-2023.)
((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) β†’ (𝐸[:]𝐾) = ((𝐸[:]𝐹) Β·e (𝐹[:]𝐾)))
 
Theoremfinexttrb 32122 The extension 𝐸 of 𝐾 is finite if and only if 𝐸 is finite over 𝐹 and 𝐹 is finite over 𝐾. Corollary 1.3 of [Lang] , p. 225. (Contributed by Thierry Arnoux, 30-Jul-2023.)
((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) β†’ (𝐸/FinExt𝐾 ↔ (𝐸/FinExt𝐹 ∧ 𝐹/FinExt𝐾)))
 
Theoremextdg1id 32123 If the degree of the extension 𝐸/FldExt𝐹 is 1, then 𝐸 and 𝐹 are identical. (Contributed by Thierry Arnoux, 6-Aug-2023.)
((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) β†’ 𝐸 = 𝐹)
 
Theoremextdg1b 32124 The degree of the extension 𝐸/FldExt𝐹 is 1 iff 𝐸 and 𝐹 are the same structure. (Contributed by Thierry Arnoux, 6-Aug-2023.)
(𝐸/FldExt𝐹 β†’ ((𝐸[:]𝐹) = 1 ↔ 𝐸 = 𝐹))
 
Theoremfldextchr 32125 The characteristic of a subfield is the same as the characteristic of the larger field. (Contributed by Thierry Arnoux, 20-Aug-2023.)
(𝐸/FldExt𝐹 β†’ (chrβ€˜πΉ) = (chrβ€˜πΈ))
 
Theoremccfldsrarelvec 32126 The subring algebra of the complex numbers over the real numbers is a left vector space. (Contributed by Thierry Arnoux, 20-Aug-2023.)
((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec
 
Theoremccfldextdgrr 32127 The degree of the field extension of the complex numbers over the real numbers is 2. (Suggested by GL, 4-Aug-2023.) (Contributed by Thierry Arnoux, 20-Aug-2023.)
(β„‚fld[:]ℝfld) = 2
 
21.3.10.1  Algebraic numbers and Minimal polynomials
 
Syntaxcalgnb 32128 Extend class notation with the algebraic number builder function.
class AlgNb
 
Syntaxcminply 32129 Extend class notation with the minimal polynomial builder function.
class minPoly
 
Definitiondf-algnb 32130* Define the algebraic number builder function. This definition is similar to df-aa 25597. (Contributed by Thierry Arnoux, 19-Jan-2025.)
AlgNb = (𝑒 ∈ V, 𝑓 ∈ V ↦ βˆͺ 𝑝 ∈ (dom (𝑒 evalSub1 𝑓) βˆ– {(0gβ€˜(Poly1β€˜π‘’))})(β—‘((𝑒 evalSub1 𝑓)β€˜π‘) β€œ {(0gβ€˜π‘’)}))
 
Definitiondf-minply 32131* Define the minimal polynomial builder function . (Contributed by Thierry Arnoux, 19-Jan-2025.)
minPoly = (𝑒 ∈ V, 𝑓 ∈ V ↦ (π‘₯ ∈ (𝑒 AlgNb 𝑓) ↦ inf((β—‘((𝑒 evalSub1 𝑓)β€˜π‘) β€œ {(0gβ€˜π‘’)}), (Monic1pβ€˜π‘’), (( deg1 β€˜π‘’) ∘r < ( deg1 β€˜π‘’)))))
 
Theoremalgnbval 32132* The algebraic numbers over a field 𝐹 within a field 𝐸. That is, the numbers 𝑋 which are roots of nonzero polynomials 𝑝(𝑋) with coefficients in (Baseβ€˜πΉ). This is expressed by the idiom (β—‘(π‘‚β€˜π‘) β€œ { 0 }), which can be translated into {π‘₯ ∈ (Baseβ€˜πΈ) ∣ ((π‘‚β€˜π‘)β€˜π‘₯) = 0 } by fniniseg2 7008. (Contributed by Thierry Arnoux, 26-Jan-2025.)
𝑂 = (𝐸 evalSub1 𝐹)    &   π‘ = (0gβ€˜(Poly1β€˜πΈ))    &    0 = (0gβ€˜πΈ)    &   (πœ‘ β†’ 𝐸 ∈ Field)    &   (πœ‘ β†’ 𝐹 ∈ (SubDRingβ€˜πΈ))    β‡’   (πœ‘ β†’ (𝐸 AlgNb 𝐹) = βˆͺ 𝑝 ∈ (dom 𝑂 βˆ– {𝑍})(β—‘(π‘‚β€˜π‘) β€œ { 0 }))
 
Theoremisalgnb 32133* Property for an element 𝑋 of a field 𝐸 to be algebraic over a subfield 𝐹. (Contributed by Thierry Arnoux, 26-Jan-2025.)
𝑂 = (𝐸 evalSub1 𝐹)    &   π‘ = (0gβ€˜(Poly1β€˜πΈ))    &    0 = (0gβ€˜πΈ)    &   (πœ‘ β†’ 𝐸 ∈ Field)    &   (πœ‘ β†’ 𝐹 ∈ (SubDRingβ€˜πΈ))    &   π΅ = (Baseβ€˜πΈ)    β‡’   (πœ‘ β†’ (𝑋 ∈ (𝐸 AlgNb 𝐹) ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘ ∈ (dom 𝑂 βˆ– {𝑍})((π‘‚β€˜π‘)β€˜π‘‹) = 0 )))
 
Theoremminplyeulem 32134* An algebraic number 𝑋 over 𝐹 is a root of some monic polynomial 𝑝 with coefficients in 𝐹. (Contributed by Thierry Arnoux, 26-Jan-2025.)
𝑂 = (𝐸 evalSub1 𝐹)    &   π‘ = (0gβ€˜(Poly1β€˜πΈ))    &    0 = (0gβ€˜πΈ)    &   (πœ‘ β†’ 𝐸 ∈ Field)    &   (πœ‘ β†’ 𝐹 ∈ (SubDRingβ€˜πΈ))    &   (πœ‘ β†’ 𝑋 ∈ (𝐸 AlgNb 𝐹))    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ (Monic1pβ€˜πΈ)((π‘‚β€˜π‘)β€˜π‘‹) = 0 )
 
21.3.11  Matrices
 
21.3.11.1  Submatrices
 
Syntaxcsmat 32135 Syntax for a function generating submatrices.
class subMat1
 
Definitiondf-smat 32136* Define a function generating submatrices of an integer-indexed matrix. The function maps an index in ((1...𝑀) Γ— (1...𝑁)) into a new index in ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 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 21848. (Contributed by Thierry Arnoux, 18-Aug-2020.)
subMat1 = (π‘š ∈ V ↦ (π‘˜ ∈ β„•, 𝑙 ∈ β„• ↦ (π‘š ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < π‘˜, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))
 
Theoremsmatfval 32137* Value of the submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
((𝐾 ∈ β„• ∧ 𝐿 ∈ β„• ∧ 𝑀 ∈ 𝑉) β†’ (𝐾(subMat1β€˜π‘€)𝐿) = (𝑀 ∘ (𝑖 ∈ β„•, 𝑗 ∈ β„• ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
 
Theoremsmatrcl 32138 Closure of the rectangular submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1β€˜π΄)𝐿)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐾 ∈ (1...𝑀))    &   (πœ‘ β†’ 𝐿 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐴 ∈ (𝐡 ↑m ((1...𝑀) Γ— (1...𝑁))))    β‡’   (πœ‘ β†’ 𝑆 ∈ (𝐡 ↑m ((1...(𝑀 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))))
 
Theoremsmatlem 32139 Lemma for the next theorems. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1β€˜π΄)𝐿)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐾 ∈ (1...𝑀))    &   (πœ‘ β†’ 𝐿 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐴 ∈ (𝐡 ↑m ((1...𝑀) Γ— (1...𝑁))))    &   (πœ‘ β†’ 𝐼 ∈ β„•)    &   (πœ‘ β†’ 𝐽 ∈ β„•)    &   (πœ‘ β†’ if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝑋)    &   (πœ‘ β†’ if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = π‘Œ)    β‡’   (πœ‘ β†’ (𝐼𝑆𝐽) = (π‘‹π΄π‘Œ))
 
Theoremsmattl 32140 Entries of a submatrix, top left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1β€˜π΄)𝐿)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐾 ∈ (1...𝑀))    &   (πœ‘ β†’ 𝐿 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐴 ∈ (𝐡 ↑m ((1...𝑀) Γ— (1...𝑁))))    &   (πœ‘ β†’ 𝐼 ∈ (1..^𝐾))    &   (πœ‘ β†’ 𝐽 ∈ (1..^𝐿))    β‡’   (πœ‘ β†’ (𝐼𝑆𝐽) = (𝐼𝐴𝐽))
 
Theoremsmattr 32141 Entries of a submatrix, top right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1β€˜π΄)𝐿)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐾 ∈ (1...𝑀))    &   (πœ‘ β†’ 𝐿 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐴 ∈ (𝐡 ↑m ((1...𝑀) Γ— (1...𝑁))))    &   (πœ‘ β†’ 𝐼 ∈ (𝐾...𝑀))    &   (πœ‘ β†’ 𝐽 ∈ (1..^𝐿))    β‡’   (πœ‘ β†’ (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽))
 
Theoremsmatbl 32142 Entries of a submatrix, bottom left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1β€˜π΄)𝐿)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐾 ∈ (1...𝑀))    &   (πœ‘ β†’ 𝐿 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐴 ∈ (𝐡 ↑m ((1...𝑀) Γ— (1...𝑁))))    &   (πœ‘ β†’ 𝐼 ∈ (1..^𝐾))    &   (πœ‘ β†’ 𝐽 ∈ (𝐿...𝑁))    β‡’   (πœ‘ β†’ (𝐼𝑆𝐽) = (𝐼𝐴(𝐽 + 1)))
 
Theoremsmatbr 32143 Entries of a submatrix, bottom right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1β€˜π΄)𝐿)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐾 ∈ (1...𝑀))    &   (πœ‘ β†’ 𝐿 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐴 ∈ (𝐡 ↑m ((1...𝑀) Γ— (1...𝑁))))    &   (πœ‘ β†’ 𝐼 ∈ (𝐾...𝑀))    &   (πœ‘ β†’ 𝐽 ∈ (𝐿...𝑁))    β‡’   (πœ‘ β†’ (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴(𝐽 + 1)))
 
Theoremsmatcl 32144 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 βˆ’ 1). (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΆ = (Baseβ€˜((1...(𝑁 βˆ’ 1)) Mat 𝑅))    &   π‘† = (𝐾(subMat1β€˜π‘€)𝐿)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐾 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐿 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝑀 ∈ 𝐡)    β‡’   (πœ‘ β†’ 𝑆 ∈ 𝐢)
 
Theoremmatmpo 32145* Write a square matrix as a mapping operation. (Contributed by Thierry Arnoux, 16-Aug-2020.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    β‡’   (𝑀 ∈ 𝐡 β†’ 𝑀 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀𝑗)))
 
Theorem1smat1 32146 The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 21854. (Contributed by Thierry Arnoux, 19-Aug-2020.)
1 = (1rβ€˜((1...𝑁) Mat 𝑅))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐼 ∈ (1...𝑁))    β‡’   (πœ‘ β†’ (𝐼(subMat1β€˜ 1 )𝐼) = (1rβ€˜((1...(𝑁 βˆ’ 1)) Mat 𝑅)))
 
Theoremsubmat1n 32147 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)    &   π΅ = (Baseβ€˜π΄)    β‡’   ((𝑁 ∈ β„• ∧ 𝑀 ∈ 𝐡) β†’ (𝑁(subMat1β€˜π‘€)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)β€˜π‘€)𝑁))
 
Theoremsubmatres 32148 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)    &   π΅ = (Baseβ€˜π΄)    β‡’   ((𝑁 ∈ β„• ∧ 𝑀 ∈ 𝐡) β†’ (𝑁(subMat1β€˜π‘€)𝑁) = (𝑀 β†Ύ ((1...(𝑁 βˆ’ 1)) Γ— (1...(𝑁 βˆ’ 1)))))
 
Theoremsubmateqlem1 32149 Lemma for submateq 32151. (Contributed by Thierry Arnoux, 25-Aug-2020.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐾 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝑀 ∈ (1...(𝑁 βˆ’ 1)))    &   (πœ‘ β†’ 𝐾 ≀ 𝑀)    β‡’   (πœ‘ β†’ (𝑀 ∈ (𝐾...𝑁) ∧ (𝑀 + 1) ∈ ((1...𝑁) βˆ– {𝐾})))
 
Theoremsubmateqlem2 32150 Lemma for submateq 32151. (Contributed by Thierry Arnoux, 26-Aug-2020.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐾 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝑀 ∈ (1...(𝑁 βˆ’ 1)))    &   (πœ‘ β†’ 𝑀 < 𝐾)    β‡’   (πœ‘ β†’ (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) βˆ– {𝐾})))
 
Theoremsubmateq 32151* Sufficient condition for two submatrices to be equal. (Contributed by Thierry Arnoux, 25-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐼 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐽 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐸 ∈ 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   ((πœ‘ ∧ 𝑖 ∈ ((1...𝑁) βˆ– {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) βˆ– {𝐽})) β†’ (𝑖𝐸𝑗) = (𝑖𝐹𝑗))    β‡’   (πœ‘ β†’ (𝐼(subMat1β€˜πΈ)𝐽) = (𝐼(subMat1β€˜πΉ)𝐽))
 
Theoremsubmatminr1 32152 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)    &   π΅ = (Baseβ€˜π΄)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐼 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐽 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑀 ∈ 𝐡)    &   πΈ = (𝐼(((1...𝑁) minMatR1 𝑅)β€˜π‘€)𝐽)    β‡’   (πœ‘ β†’ (𝐼(subMat1β€˜π‘€)𝐽) = (𝐼(subMat1β€˜πΈ)𝐽))
 
21.3.11.2  Matrix literals
 
Syntaxclmat 32153 Extend class notation with the literal matrix conversion function.
class litMat
 
Definitiondf-lmat 32154* Define a function converting words of words into matrices. (Contributed by Thierry Arnoux, 28-Aug-2020.)
litMat = (π‘š ∈ V ↦ (𝑖 ∈ (1...(β™―β€˜π‘š)), 𝑗 ∈ (1...(β™―β€˜(π‘šβ€˜0))) ↦ ((π‘šβ€˜(𝑖 βˆ’ 1))β€˜(𝑗 βˆ’ 1))))
 
Theoremlmatval 32155* Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.)
(𝑀 ∈ 𝑉 β†’ (litMatβ€˜π‘€) = (𝑖 ∈ (1...(β™―β€˜π‘€)), 𝑗 ∈ (1...(β™―β€˜(π‘€β€˜0))) ↦ ((π‘€β€˜(𝑖 βˆ’ 1))β€˜(𝑗 βˆ’ 1))))
 
Theoremlmatfval 32156* Entries of a literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMatβ€˜π‘Š)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ π‘Š ∈ Word Word 𝑉)    &   (πœ‘ β†’ (β™―β€˜π‘Š) = 𝑁)    &   ((πœ‘ ∧ 𝑖 ∈ (0..^𝑁)) β†’ (β™―β€˜(π‘Šβ€˜π‘–)) = 𝑁)    &   (πœ‘ β†’ 𝐼 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐽 ∈ (1...𝑁))    β‡’   (πœ‘ β†’ (𝐼𝑀𝐽) = ((π‘Šβ€˜(𝐼 βˆ’ 1))β€˜(𝐽 βˆ’ 1)))
 
Theoremlmatfvlem 32157* Useful lemma to extract literal matrix entries. Suggested by Mario Carneiro. (Contributed by Thierry Arnoux, 3-Sep-2020.)
𝑀 = (litMatβ€˜π‘Š)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ π‘Š ∈ Word Word 𝑉)    &   (πœ‘ β†’ (β™―β€˜π‘Š) = 𝑁)    &   ((πœ‘ ∧ 𝑖 ∈ (0..^𝑁)) β†’ (β™―β€˜(π‘Šβ€˜π‘–)) = 𝑁)    &   πΎ ∈ β„•0    &   πΏ ∈ β„•0    &   πΌ ≀ 𝑁    &   π½ ≀ 𝑁    &   (𝐾 + 1) = 𝐼    &   (𝐿 + 1) = 𝐽    &   (π‘Šβ€˜πΎ) = 𝑋    &   (πœ‘ β†’ (π‘‹β€˜πΏ) = π‘Œ)    β‡’   (πœ‘ β†’ (𝐼𝑀𝐽) = π‘Œ)
 
Theoremlmatcl 32158* Closure of the literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMatβ€˜π‘Š)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ π‘Š ∈ Word Word 𝑉)    &   (πœ‘ β†’ (β™―β€˜π‘Š) = 𝑁)    &   ((πœ‘ ∧ 𝑖 ∈ (0..^𝑁)) β†’ (β™―β€˜(π‘Šβ€˜π‘–)) = 𝑁)    &   π‘‰ = (Baseβ€˜π‘…)    &   π‘‚ = ((1...𝑁) Mat 𝑅)    &   π‘ƒ = (Baseβ€˜π‘‚)    &   (πœ‘ β†’ 𝑅 ∈ 𝑋)    β‡’   (πœ‘ β†’ 𝑀 ∈ 𝑃)
 
Theoremlmat22lem 32159* Lemma for lmat22e11 32160 and co. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMatβ€˜βŸ¨β€œβŸ¨β€œπ΄π΅β€βŸ©βŸ¨β€œπΆπ·β€βŸ©β€βŸ©)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ 𝑉)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    β‡’   ((πœ‘ ∧ 𝑖 ∈ (0..^2)) β†’ (β™―β€˜(βŸ¨β€œβŸ¨β€œπ΄π΅β€βŸ©βŸ¨β€œπΆπ·β€βŸ©β€βŸ©β€˜π‘–)) = 2)
 
Theoremlmat22e11 32160 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMatβ€˜βŸ¨β€œβŸ¨β€œπ΄π΅β€βŸ©βŸ¨β€œπΆπ·β€βŸ©β€βŸ©)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ 𝑉)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    β‡’   (πœ‘ β†’ (1𝑀1) = 𝐴)
 
Theoremlmat22e12 32161 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMatβ€˜βŸ¨β€œβŸ¨β€œπ΄π΅β€βŸ©βŸ¨β€œπΆπ·β€βŸ©β€βŸ©)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ 𝑉)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    β‡’   (πœ‘ β†’ (1𝑀2) = 𝐡)
 
Theoremlmat22e21 32162 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMatβ€˜βŸ¨β€œβŸ¨β€œπ΄π΅β€βŸ©βŸ¨β€œπΆπ·β€βŸ©β€βŸ©)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ 𝑉)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    β‡’   (πœ‘ β†’ (2𝑀1) = 𝐢)
 
Theoremlmat22e22 32163 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMatβ€˜βŸ¨β€œβŸ¨β€œπ΄π΅β€βŸ©βŸ¨β€œπΆπ·β€βŸ©β€βŸ©)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ 𝑉)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    β‡’   (πœ‘ β†’ (2𝑀2) = 𝐷)
 
Theoremlmat22det 32164 The determinant of a literal 2x2 complex matrix. (Contributed by Thierry Arnoux, 1-Sep-2020.)
𝑀 = (litMatβ€˜βŸ¨β€œβŸ¨β€œπ΄π΅β€βŸ©βŸ¨β€œπΆπ·β€βŸ©β€βŸ©)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ 𝑉)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &    Β· = (.rβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘…)    &   π‘‰ = (Baseβ€˜π‘…)    &   π½ = ((1...2) maDet 𝑅)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ (π½β€˜π‘€) = ((𝐴 Β· 𝐷) βˆ’ (𝐢 Β· 𝐡)))
 
21.3.11.3  Laplace expansion of determinants
 
Theoremmdetpmtr1 32165* 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)    &   π΅ = (Baseβ€˜π΄)    &   π· = (𝑁 maDet 𝑅)    &   πΊ = (Baseβ€˜(SymGrpβ€˜π‘))    &   π‘† = (pmSgnβ€˜π‘)    &   π‘ = (β„€RHomβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   πΈ = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((π‘ƒβ€˜π‘–)𝑀𝑗))    β‡’   (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐡 ∧ 𝑃 ∈ 𝐺)) β†’ (π·β€˜π‘€) = (((𝑍 ∘ 𝑆)β€˜π‘ƒ) Β· (π·β€˜πΈ)))
 
Theoremmdetpmtr2 32166* 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)    &   π΅ = (Baseβ€˜π΄)    &   π· = (𝑁 maDet 𝑅)    &   πΊ = (Baseβ€˜(SymGrpβ€˜π‘))    &   π‘† = (pmSgnβ€˜π‘)    &   π‘ = (β„€RHomβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   πΈ = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(π‘ƒβ€˜π‘—)))    β‡’   (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐡 ∧ 𝑃 ∈ 𝐺)) β†’ (π·β€˜π‘€) = (((𝑍 ∘ 𝑆)β€˜π‘ƒ) Β· (π·β€˜πΈ)))
 
Theoremmdetpmtr12 32167* 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)    &   π΅ = (Baseβ€˜π΄)    &   π· = (𝑁 maDet 𝑅)    &   πΊ = (Baseβ€˜(SymGrpβ€˜π‘))    &   π‘† = (pmSgnβ€˜π‘)    &   π‘ = (β„€RHomβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   πΈ = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((π‘ƒβ€˜π‘–)𝑀(π‘„β€˜π‘—)))    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ 𝑀 ∈ 𝐡)    &   (πœ‘ β†’ 𝑃 ∈ 𝐺)    &   (πœ‘ β†’ 𝑄 ∈ 𝐺)    β‡’   (πœ‘ β†’ (π·β€˜π‘€) = ((π‘β€˜((π‘†β€˜π‘ƒ) Β· (π‘†β€˜π‘„))) Β· (π·β€˜πΈ)))
 
Theoremmdetlap1 32168* 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)    &   π΅ = (Baseβ€˜π΄)    &   π· = (𝑁 maDet 𝑅)    &   πΎ = (𝑁 maAdju 𝑅)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡 ∧ 𝐼 ∈ 𝑁) β†’ (π·β€˜π‘€) = (𝑅 Ξ£g (𝑗 ∈ 𝑁 ↦ ((𝐼𝑀𝑗) Β· (𝑗(πΎβ€˜π‘€)𝐼)))))
 
Theoremmadjusmdetlem1 32169* Lemma for madjusmdet 32173. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐡 = (Baseβ€˜π΄)    &   π΄ = ((1...𝑁) Mat 𝑅)    &   π· = ((1...𝑁) maDet 𝑅)    &   πΎ = ((1...𝑁) maAdju 𝑅)    &    Β· = (.rβ€˜π‘…)    &   π‘ = (β„€RHomβ€˜π‘…)    &   πΈ = ((1...(𝑁 βˆ’ 1)) maDet 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐼 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐽 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝑀 ∈ 𝐡)    &   πΊ = (Baseβ€˜(SymGrpβ€˜(1...𝑁)))    &   π‘† = (pmSgnβ€˜(1...𝑁))    &   π‘ˆ = (𝐼(((1...𝑁) minMatR1 𝑅)β€˜π‘€)𝐽)    &   π‘Š = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((π‘ƒβ€˜π‘–)π‘ˆ(π‘„β€˜π‘—)))    &   (πœ‘ β†’ 𝑃 ∈ 𝐺)    &   (πœ‘ β†’ 𝑄 ∈ 𝐺)    &   (πœ‘ β†’ (π‘ƒβ€˜π‘) = 𝐼)    &   (πœ‘ β†’ (π‘„β€˜π‘) = 𝐽)    &   (πœ‘ β†’ (𝐼(subMat1β€˜π‘ˆ)𝐽) = (𝑁(subMat1β€˜π‘Š)𝑁))    β‡’   (πœ‘ β†’ (𝐽(πΎβ€˜π‘€)𝐼) = ((π‘β€˜((π‘†β€˜π‘ƒ) Β· (π‘†β€˜π‘„))) Β· (πΈβ€˜(𝐼(subMat1β€˜π‘€)𝐽))))
 
Theoremmadjusmdetlem2 32170* Lemma for madjusmdet 32173. (Contributed by Thierry Arnoux, 26-Aug-2020.)
𝐡 = (Baseβ€˜π΄)    &   π΄ = ((1...𝑁) Mat 𝑅)    &   π· = ((1...𝑁) maDet 𝑅)    &   πΎ = ((1...𝑁) maAdju 𝑅)    &    Β· = (.rβ€˜π‘…)    &   π‘ = (β„€RHomβ€˜π‘…)    &   πΈ = ((1...(𝑁 βˆ’ 1)) maDet 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐼 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐽 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝑀 ∈ 𝐡)    &   π‘ƒ = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≀ 𝐼, (𝑖 βˆ’ 1), 𝑖)))    &   π‘† = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≀ 𝑁, (𝑖 βˆ’ 1), 𝑖)))    β‡’   ((πœ‘ ∧ 𝑋 ∈ (1...(𝑁 βˆ’ 1))) β†’ if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃 ∘ ◑𝑆)β€˜π‘‹))
 
Theoremmadjusmdetlem3 32171* Lemma for madjusmdet 32173. (Contributed by Thierry Arnoux, 27-Aug-2020.)
𝐡 = (Baseβ€˜π΄)    &   π΄ = ((1...𝑁) Mat 𝑅)    &   π· = ((1...𝑁) maDet 𝑅)    &   πΎ = ((1...𝑁) maAdju 𝑅)    &    Β· = (.rβ€˜π‘…)    &   π‘ = (β„€RHomβ€˜π‘…)    &   πΈ = ((1...(𝑁 βˆ’ 1)) maDet 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐼 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐽 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝑀 ∈ 𝐡)    &   π‘ƒ = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≀ 𝐼, (𝑖 βˆ’ 1), 𝑖)))    &   π‘† = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≀ 𝑁, (𝑖 βˆ’ 1), 𝑖)))    &   π‘„ = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≀ 𝐽, (𝑗 βˆ’ 1), 𝑗)))    &   π‘‡ = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≀ 𝑁, (𝑗 βˆ’ 1), 𝑗)))    &   π‘Š = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ◑𝑆)β€˜π‘–)π‘ˆ((𝑄 ∘ ◑𝑇)β€˜π‘—)))    &   (πœ‘ β†’ π‘ˆ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐼(subMat1β€˜π‘ˆ)𝐽) = (𝑁(subMat1β€˜π‘Š)𝑁))
 
Theoremmadjusmdetlem4 32172* Lemma for madjusmdet 32173. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐡 = (Baseβ€˜π΄)    &   π΄ = ((1...𝑁) Mat 𝑅)    &   π· = ((1...𝑁) maDet 𝑅)    &   πΎ = ((1...𝑁) maAdju 𝑅)    &    Β· = (.rβ€˜π‘…)    &   π‘ = (β„€RHomβ€˜π‘…)    &   πΈ = ((1...(𝑁 βˆ’ 1)) maDet 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐼 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐽 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝑀 ∈ 𝐡)    &   π‘ƒ = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≀ 𝐼, (𝑖 βˆ’ 1), 𝑖)))    &   π‘† = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≀ 𝑁, (𝑖 βˆ’ 1), 𝑖)))    &   π‘„ = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≀ 𝐽, (𝑗 βˆ’ 1), 𝑗)))    &   π‘‡ = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≀ 𝑁, (𝑗 βˆ’ 1), 𝑗)))    β‡’   (πœ‘ β†’ (𝐽(πΎβ€˜π‘€)𝐼) = ((π‘β€˜(-1↑(𝐼 + 𝐽))) Β· (πΈβ€˜(𝐼(subMat1β€˜π‘€)𝐽))))
 
Theoremmadjusmdet 32173 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.)
𝐡 = (Baseβ€˜π΄)    &   π΄ = ((1...𝑁) Mat 𝑅)    &   π· = ((1...𝑁) maDet 𝑅)    &   πΎ = ((1...𝑁) maAdju 𝑅)    &    Β· = (.rβ€˜π‘…)    &   π‘ = (β„€RHomβ€˜π‘…)    &   πΈ = ((1...(𝑁 βˆ’ 1)) maDet 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐼 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐽 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝑀 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐽(πΎβ€˜π‘€)𝐼) = ((π‘β€˜(-1↑(𝐼 + 𝐽))) Β· (πΈβ€˜(𝐼(subMat1β€˜π‘€)𝐽))))
 
Theoremmdetlap 32174* Laplace expansion of the determinant of a square matrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝐡 = (Baseβ€˜π΄)    &   π΄ = ((1...𝑁) Mat 𝑅)    &   π· = ((1...𝑁) maDet 𝑅)    &   πΎ = ((1...𝑁) maAdju 𝑅)    &    Β· = (.rβ€˜π‘…)    &   π‘ = (β„€RHomβ€˜π‘…)    &   πΈ = ((1...(𝑁 βˆ’ 1)) maDet 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐼 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐽 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝑀 ∈ 𝐡)    β‡’   (πœ‘ β†’ (π·β€˜π‘€) = (𝑅 Ξ£g (𝑗 ∈ (1...𝑁) ↦ ((π‘β€˜(-1↑(𝐼 + 𝑗))) Β· ((𝐼𝑀𝑗) Β· (πΈβ€˜(𝐼(subMat1β€˜π‘€)𝑗)))))))
 
21.3.12  Topology
 
Theoremist0cld 32175* The predicate "is a T0 space", using closed sets. (Contributed by Thierry Arnoux, 16-Aug-2020.)
(πœ‘ β†’ 𝐡 = βˆͺ 𝐽)    &   (πœ‘ β†’ 𝐷 = (Clsdβ€˜π½))    β‡’   (πœ‘ β†’ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (βˆ€π‘‘ ∈ 𝐷 (π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) β†’ π‘₯ = 𝑦))))
 
21.3.12.1  Open maps
 
Theoremtxomap 32176* 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.)
(πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘)    &   (πœ‘ β†’ 𝐺:π‘ŒβŸΆπ‘‡)    &   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))    &   (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜π‘‡))    &   ((πœ‘ ∧ π‘₯ ∈ 𝐽) β†’ (𝐹 β€œ π‘₯) ∈ 𝐿)    &   ((πœ‘ ∧ 𝑦 ∈ 𝐾) β†’ (𝐺 β€œ 𝑦) ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ (𝐽 Γ—t 𝐾))    &   π» = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ ⟨(πΉβ€˜π‘₯), (πΊβ€˜π‘¦)⟩)    β‡’   (πœ‘ β†’ (𝐻 β€œ 𝐴) ∈ (𝐿 Γ—t 𝑀))
 
21.3.12.2  Topology of the unit circle
 
Theoremqtopt1 32177* If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.)
𝑋 = βˆͺ 𝐽    &   (πœ‘ β†’ 𝐽 ∈ Fre)    &   (πœ‘ β†’ 𝐹:𝑋–ontoβ†’π‘Œ)    &   ((πœ‘ ∧ π‘₯ ∈ π‘Œ) β†’ (◑𝐹 β€œ {π‘₯}) ∈ (Clsdβ€˜π½))    β‡’   (πœ‘ β†’ (𝐽 qTop 𝐹) ∈ Fre)
 
Theoremqtophaus 32178* If an open map's graph in the product space (𝐽 Γ—t 𝐽) is closed, then its quotient topology is Hausdorff. (Contributed by Thierry Arnoux, 4-Jan-2020.)
𝑋 = βˆͺ 𝐽    &    ∼ = (◑𝐹 ∘ 𝐹)    &   π» = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(πΉβ€˜π‘₯), (πΉβ€˜π‘¦)⟩)    &   (πœ‘ β†’ 𝐽 ∈ Haus)    &   (πœ‘ β†’ 𝐹:𝑋–ontoβ†’π‘Œ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐽) β†’ (𝐹 β€œ π‘₯) ∈ (𝐽 qTop 𝐹))    &   (πœ‘ β†’ ∼ ∈ (Clsdβ€˜(𝐽 Γ—t 𝐽)))    β‡’   (πœ‘ β†’ (𝐽 qTop 𝐹) ∈ Haus)
 
Theoremcirctopn 32179* 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 Β· Ο€))    &   π½ = (topGenβ€˜ran (,))    &   πΉ = (π‘₯ ∈ ℝ ↦ (expβ€˜(i Β· π‘₯)))    &   πΆ = (β—‘abs β€œ {1})    β‡’   (𝐽 qTop 𝐹) = (TopOpenβ€˜(𝐹 β€œs ℝfld))
 
Theoremcirccn 32180* The function gluing the real line into the unit circle is continuous. (Contributed by Thierry Arnoux, 5-Jan-2020.)
𝐼 = (0[,](2 Β· Ο€))    &   π½ = (topGenβ€˜ran (,))    &   πΉ = (π‘₯ ∈ ℝ ↦ (expβ€˜(i Β· π‘₯)))    &   πΆ = (β—‘abs β€œ {1})    β‡’   πΉ ∈ (𝐽 Cn (𝐽 qTop 𝐹))
 
21.3.12.3  Refinements
 
Theoremreff 32181* 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 10358. (Contributed by Thierry Arnoux, 12-Jan-2020.)
(𝐴 ∈ 𝑉 β†’ (𝐴Ref𝐡 ↔ (βˆͺ 𝐡 βŠ† βˆͺ 𝐴 ∧ βˆƒπ‘“(𝑓:𝐴⟢𝐡 ∧ βˆ€π‘£ ∈ 𝐴 𝑣 βŠ† (π‘“β€˜π‘£)))))
 
Theoremlocfinreflem 32182* 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 π‘ˆ, 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 = βˆͺ 𝐽    &   (πœ‘ β†’ π‘ˆ βŠ† 𝐽)    &   (πœ‘ β†’ 𝑋 = βˆͺ π‘ˆ)    &   (πœ‘ β†’ 𝑉 βŠ† 𝐽)    &   (πœ‘ β†’ 𝑉Refπ‘ˆ)    &   (πœ‘ β†’ 𝑉 ∈ (LocFinβ€˜π½))    β‡’   (πœ‘ β†’ βˆƒπ‘“((Fun 𝑓 ∧ dom 𝑓 βŠ† π‘ˆ ∧ ran 𝑓 βŠ† 𝐽) ∧ (ran 𝑓Refπ‘ˆ ∧ ran 𝑓 ∈ (LocFinβ€˜π½))))
 
Theoremlocfinref 32183* 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 π‘ˆ, which is taken as the index set. (Contributed by Thierry Arnoux, 31-Jan-2020.)
𝑋 = βˆͺ 𝐽    &   (πœ‘ β†’ π‘ˆ βŠ† 𝐽)    &   (πœ‘ β†’ 𝑋 = βˆͺ π‘ˆ)    &   (πœ‘ β†’ 𝑉 βŠ† 𝐽)    &   (πœ‘ β†’ 𝑉Refπ‘ˆ)    &   (πœ‘ β†’ 𝑉 ∈ (LocFinβ€˜π½))    β‡’   (πœ‘ β†’ βˆƒπ‘“(𝑓:π‘ˆβŸΆπ½ ∧ ran 𝑓Refπ‘ˆ ∧ ran 𝑓 ∈ (LocFinβ€˜π½)))
 
21.3.12.4  Open cover refinement property
 
Syntaxccref 32184 The "every open cover has an 𝐴 refinement" predicate.
class CovHasRef𝐴
 
Definitiondf-cref 32185* 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 ∣ βˆ€π‘¦ ∈ 𝒫 𝑗(βˆͺ 𝑗 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
 
Theoremiscref 32186* The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = βˆͺ 𝐽    β‡’   (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ βˆ€π‘¦ ∈ 𝒫 𝐽(𝑋 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
 
Theoremcrefeq 32187 Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴 = 𝐡 β†’ CovHasRef𝐴 = CovHasRef𝐡)
 
Theoremcreftop 32188 A space where every open cover has an 𝐴 refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ CovHasRef𝐴 β†’ 𝐽 ∈ Top)
 
Theoremcrefi 32189* The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐢 βŠ† 𝐽 ∧ 𝑋 = βˆͺ 𝐢) β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐢)
 
Theoremcrefdf 32190* A formulation of crefi 32189 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = βˆͺ 𝐽    &   π΅ = CovHasRef𝐴    &   (𝑧 ∈ 𝐴 β†’ πœ‘)    β‡’   ((𝐽 ∈ 𝐡 ∧ 𝐢 βŠ† 𝐽 ∧ 𝑋 = βˆͺ 𝐢) β†’ βˆƒπ‘§ ∈ 𝒫 𝐽(πœ‘ ∧ 𝑧Ref𝐢))
 
Theoremcrefss 32191 The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴 βŠ† 𝐡 β†’ CovHasRef𝐴 βŠ† CovHasRef𝐡)
 
Theoremcmpcref 32192 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 32193* Every open cover of a Compact space has a finite refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Comp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆƒπ‘£ ∈ 𝒫 𝐽(𝑣 ∈ Fin ∧ 𝑣Refπ‘ˆ))
 
21.3.12.5  LindelΓΆf spaces
 
Syntaxcldlf 32194 Extend class notation with the class of all LindelΓΆf spaces.
class Ldlf
 
Definitiondf-ldlf 32195 Definition of a LindelΓΆf space. A LindelΓΆf 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{π‘₯ ∣ π‘₯ β‰Ό Ο‰}
 
Theoremldlfcntref 32196* Every open cover of a LindelΓΆf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Ldlf ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆƒπ‘£ ∈ 𝒫 𝐽(𝑣 β‰Ό Ο‰ ∧ 𝑣Refπ‘ˆ))
 
21.3.12.6  Paracompact spaces
 
Syntaxcpcmp 32197 Extend class notation with the class of all paracompact topologies.
class Paracomp
 
Definitiondf-pcmp 32198 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(LocFinβ€˜π‘—)}
 
Theoremispcmp 32199 The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFinβ€˜π½))
 
Theoremcmppcmp 32200 Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Comp β†’ 𝐽 ∈ Paracomp)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46948
  Copyright terms: Public domain < Previous  Next >