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Theorem List for Metamath Proof Explorer - 45501-45600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembgoldbtbndlem1 45501 Lemma 1 for bgoldbtbnd 45505: the odd numbers between 7 and 13 (exclusive) are odd Goldbach numbers. (Contributed by AV, 29-Jul-2020.)
((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 ∈ (7[,)13)) β†’ 𝑁 ∈ GoldbachOdd )
 
Theorembgoldbtbndlem2 45502* Lemma 2 for bgoldbtbnd 45505. (Contributed by AV, 1-Aug-2020.)
(πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜11))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜11))    &   (πœ‘ β†’ βˆ€π‘› ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑁) β†’ 𝑛 ∈ GoldbachEven ))    &   (πœ‘ β†’ 𝐷 ∈ (β„€β‰₯β€˜3))    &   (πœ‘ β†’ 𝐹 ∈ (RePartβ€˜π·))    &   (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝐷)((πΉβ€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((πΉβ€˜(𝑖 + 1)) βˆ’ (πΉβ€˜π‘–)) < (𝑁 βˆ’ 4) ∧ 4 < ((πΉβ€˜(𝑖 + 1)) βˆ’ (πΉβ€˜π‘–))))    &   (πœ‘ β†’ (πΉβ€˜0) = 7)    &   (πœ‘ β†’ (πΉβ€˜1) = 13)    &   (πœ‘ β†’ 𝑀 < (πΉβ€˜π·))    &   π‘† = (𝑋 βˆ’ (πΉβ€˜(𝐼 βˆ’ 1)))    β‡’   ((πœ‘ ∧ 𝑋 ∈ Odd ∧ 𝐼 ∈ (1..^𝐷)) β†’ ((𝑋 ∈ ((πΉβ€˜πΌ)[,)(πΉβ€˜(𝐼 + 1))) ∧ (𝑋 βˆ’ (πΉβ€˜πΌ)) ≀ 4) β†’ (𝑆 ∈ Even ∧ 𝑆 < 𝑁 ∧ 4 < 𝑆)))
 
Theorembgoldbtbndlem3 45503* Lemma 3 for bgoldbtbnd 45505. (Contributed by AV, 1-Aug-2020.)
(πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜11))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜11))    &   (πœ‘ β†’ βˆ€π‘› ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑁) β†’ 𝑛 ∈ GoldbachEven ))    &   (πœ‘ β†’ 𝐷 ∈ (β„€β‰₯β€˜3))    &   (πœ‘ β†’ 𝐹 ∈ (RePartβ€˜π·))    &   (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝐷)((πΉβ€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((πΉβ€˜(𝑖 + 1)) βˆ’ (πΉβ€˜π‘–)) < (𝑁 βˆ’ 4) ∧ 4 < ((πΉβ€˜(𝑖 + 1)) βˆ’ (πΉβ€˜π‘–))))    &   (πœ‘ β†’ (πΉβ€˜0) = 7)    &   (πœ‘ β†’ (πΉβ€˜1) = 13)    &   (πœ‘ β†’ 𝑀 < (πΉβ€˜π·))    &   (πœ‘ β†’ (πΉβ€˜π·) ∈ ℝ)    &   π‘† = (𝑋 βˆ’ (πΉβ€˜πΌ))    β‡’   ((πœ‘ ∧ 𝑋 ∈ Odd ∧ 𝐼 ∈ (1..^𝐷)) β†’ ((𝑋 ∈ ((πΉβ€˜πΌ)[,)(πΉβ€˜(𝐼 + 1))) ∧ 4 < 𝑆) β†’ (𝑆 ∈ Even ∧ 𝑆 < 𝑁 ∧ 4 < 𝑆)))
 
Theorembgoldbtbndlem4 45504* Lemma 4 for bgoldbtbnd 45505. (Contributed by AV, 1-Aug-2020.)
(πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜11))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜11))    &   (πœ‘ β†’ βˆ€π‘› ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑁) β†’ 𝑛 ∈ GoldbachEven ))    &   (πœ‘ β†’ 𝐷 ∈ (β„€β‰₯β€˜3))    &   (πœ‘ β†’ 𝐹 ∈ (RePartβ€˜π·))    &   (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝐷)((πΉβ€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((πΉβ€˜(𝑖 + 1)) βˆ’ (πΉβ€˜π‘–)) < (𝑁 βˆ’ 4) ∧ 4 < ((πΉβ€˜(𝑖 + 1)) βˆ’ (πΉβ€˜π‘–))))    &   (πœ‘ β†’ (πΉβ€˜0) = 7)    &   (πœ‘ β†’ (πΉβ€˜1) = 13)    &   (πœ‘ β†’ 𝑀 < (πΉβ€˜π·))    &   (πœ‘ β†’ (πΉβ€˜π·) ∈ ℝ)    β‡’   (((πœ‘ ∧ 𝐼 ∈ (1..^𝐷)) ∧ 𝑋 ∈ Odd ) β†’ ((𝑋 ∈ ((πΉβ€˜πΌ)[,)(πΉβ€˜(𝐼 + 1))) ∧ (𝑋 βˆ’ (πΉβ€˜πΌ)) ≀ 4) β†’ βˆƒπ‘ ∈ β„™ βˆƒπ‘ž ∈ β„™ βˆƒπ‘Ÿ ∈ β„™ ((𝑝 ∈ Odd ∧ π‘ž ∈ Odd ∧ π‘Ÿ ∈ Odd ) ∧ 𝑋 = ((𝑝 + π‘ž) + π‘Ÿ))))
 
Theorembgoldbtbnd 45505* If the binary Goldbach conjecture is valid up to an integer 𝑁, and there is a series ("ladder") of primes with a difference of at most 𝑁 up to an integer 𝑀, then the strong ternary Goldbach conjecture is valid up to 𝑀, see section 1.2.2 in [Helfgott] p. 4 with N = 4 x 10^18, taken from [OeSilva], and M = 8.875 x 10^30. (Contributed by AV, 1-Aug-2020.)
(πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜11))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜11))    &   (πœ‘ β†’ βˆ€π‘› ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑁) β†’ 𝑛 ∈ GoldbachEven ))    &   (πœ‘ β†’ 𝐷 ∈ (β„€β‰₯β€˜3))    &   (πœ‘ β†’ 𝐹 ∈ (RePartβ€˜π·))    &   (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝐷)((πΉβ€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((πΉβ€˜(𝑖 + 1)) βˆ’ (πΉβ€˜π‘–)) < (𝑁 βˆ’ 4) ∧ 4 < ((πΉβ€˜(𝑖 + 1)) βˆ’ (πΉβ€˜π‘–))))    &   (πœ‘ β†’ (πΉβ€˜0) = 7)    &   (πœ‘ β†’ (πΉβ€˜1) = 13)    &   (πœ‘ β†’ 𝑀 < (πΉβ€˜π·))    &   (πœ‘ β†’ (πΉβ€˜π·) ∈ ℝ)    β‡’   (πœ‘ β†’ βˆ€π‘› ∈ Odd ((7 < 𝑛 ∧ 𝑛 < 𝑀) β†’ 𝑛 ∈ GoldbachOdd ))
 
Axiomax-bgbltosilva 45506 The binary Goldbach conjecture is valid for all even numbers less than or equal to 4x10^18, see section 2 in [OeSilva] p. 2042. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)
((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 ≀ (4 Β· (10↑18))) β†’ 𝑁 ∈ GoldbachEven )
 
Axiomax-tgoldbachgt 45507* Temporary duplicate of tgoldbachgt 32692, provided as "axiom" as long as this theorem is in the mathbox of Thierry Arnoux: Odd integers greater than (10↑27) have at least a representation as a sum of three odd primes. Final statement in section 7.4 of [Helfgott] p. 70 , expressed using the set 𝐺 of odd numbers which can be written as a sum of three odd primes. (Contributed by Thierry Arnoux, 22-Dec-2021.)
𝑂 = {𝑧 ∈ β„€ ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΊ = {𝑧 ∈ 𝑂 ∣ βˆƒπ‘ ∈ β„™ βˆƒπ‘ž ∈ β„™ βˆƒπ‘Ÿ ∈ β„™ ((𝑝 ∈ 𝑂 ∧ π‘ž ∈ 𝑂 ∧ π‘Ÿ ∈ 𝑂) ∧ 𝑧 = ((𝑝 + π‘ž) + π‘Ÿ))}    β‡’   βˆƒπ‘š ∈ β„• (π‘š ≀ (10↑27) ∧ βˆ€π‘› ∈ 𝑂 (π‘š < 𝑛 β†’ 𝑛 ∈ 𝐺))
 
TheoremtgoldbachgtALTV 45508* Variant of Thierry Arnoux's tgoldbachgt 32692 using the symbols Odd and GoldbachOdd: The ternary Goldbach conjecture is valid for large odd numbers (i.e. for all odd numbers greater than a fixed π‘š). This is proven by Helfgott (see section 7.4 in [Helfgott] p. 70) for π‘š = 10^27. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 15-Jan-2022.)
βˆƒπ‘š ∈ β„• (π‘š ≀ (10↑27) ∧ βˆ€π‘› ∈ Odd (π‘š < 𝑛 β†’ 𝑛 ∈ GoldbachOdd ))
 
Theorembgoldbachlt 45509* The binary Goldbach conjecture is valid for small even numbers (i.e. for all even numbers less than or equal to a fixed big π‘š). This is verified for m = 4 x 10^18 by Oliveira e Silva, see ax-bgbltosilva 45506. (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)
βˆƒπ‘š ∈ β„• ((4 Β· (10↑18)) ≀ π‘š ∧ βˆ€π‘› ∈ Even ((4 < 𝑛 ∧ 𝑛 < π‘š) β†’ 𝑛 ∈ GoldbachEven ))
 
Axiomax-hgprmladder 45510 There is a partition ("ladder") of primes from 7 to 8.8 x 10^30 with parts ("rungs") having lengths of at least 4 and at most N - 4, see section 1.2.2 in [Helfgott] p. 4. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)
βˆƒπ‘‘ ∈ (β„€β‰₯β€˜3)βˆƒπ‘“ ∈ (RePartβ€˜π‘‘)(((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))
 
Theoremtgblthelfgott 45511 The ternary Goldbach conjecture is valid for all odd numbers less than 8.8 x 10^30 (actually 8.875694 x 10^30, see section 1.2.2 in [Helfgott] p. 4, using bgoldbachlt 45509, ax-hgprmladder 45510 and bgoldbtbnd 45505. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29))) β†’ 𝑁 ∈ GoldbachOdd )
 
Theoremtgoldbachlt 45512* The ternary Goldbach conjecture is valid for small odd numbers (i.e. for all odd numbers less than a fixed big π‘š greater than 8 x 10^30). This is verified for m = 8.875694 x 10^30 by Helfgott, see tgblthelfgott 45511. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
βˆƒπ‘š ∈ β„• ((8 Β· (10↑30)) < π‘š ∧ βˆ€π‘› ∈ Odd ((7 < 𝑛 ∧ 𝑛 < π‘š) β†’ 𝑛 ∈ GoldbachOdd ))
 
Theoremtgoldbach 45513 The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 45512 and ax-tgoldbachgt 45507. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.)
βˆ€π‘› ∈ Odd (7 < 𝑛 β†’ 𝑛 ∈ GoldbachOdd )
 
20.41.15  Graph theory (extension)
 
20.41.15.1  Isomorphic graphs

In the following, a general definition of the isomorphy relation for graphs and specializations for simple hypergraphs (isomushgr 45522) and simple pseudographs (isomuspgr 45530) are provided. The latter corresponds to the definition in [Bollobas] p. 3). It is shown that the isomorphy relation for graphs is an equivalence relation (isomgrref 45531, isomgrsym 45532, isomgrtr 45535). Fianlly, isomorphic graphs with different representations are studied (strisomgrop 45536, ushrisomgr 45537).

Maybe more important than graph isomorphy is the notion of graph isomorphism, which can be defined as in df-grisom 45516. Then 𝐴 IsomGr 𝐡 ↔ βˆƒπ‘“π‘“ ∈ (𝐴 GrIsom 𝐡) resp. 𝐴 IsomGr 𝐡 ↔ (𝐴 GrIsom 𝐡) β‰  βˆ…. Notice that there can be multiple isomorphisms between two graphs (let ⟨{𝐴, 𝐡}, {{𝐴, 𝐡}}⟩ and ⟨{{𝑀, 𝑁}, {{𝑀, 𝑁}}⟩ be two graphs with two vertices and one edge, then 𝐴 ↦ 𝑀, 𝐡 ↦ 𝑁 and 𝐴 ↦ 𝑁, 𝐡 ↦ 𝑀 are two different isomorphisms between these graphs).

Another approach could be to define a category of graphs (there are maybe multiple ones), where graph morphisms are couples consisting in a function on vertices and a function on edges with required compatibilities, as used in the definition of GrIsom. And then, a graph isomorphism is defined as an isomorphism in the category of graphs (something like "GraphIsom = ( Iso ` GraphCat )" ). Then general category theory theorems could be used, e.g., to show that graph isomorphy is an equivalence relation.

 
Syntaxcgrisom 45514 Extend class notation to include the graph ispmorphisms.
class GrIsom
 
Syntaxcisomgr 45515 Extend class notation to include the isomorphy relation for graphs.
class IsomGr
 
Definitiondf-grisom 45516* Define the class of all isomorphisms between two graphs. (Contributed by AV, 11-Dec-2022.)
GrIsom = (π‘₯ ∈ V, 𝑦 ∈ V ↦ {βŸ¨π‘“, π‘”βŸ© ∣ (𝑓:(Vtxβ€˜π‘₯)–1-1-ontoβ†’(Vtxβ€˜π‘¦) ∧ 𝑔:dom (iEdgβ€˜π‘₯)–1-1-ontoβ†’dom (iEdgβ€˜π‘¦) ∧ βˆ€π‘– ∈ dom (iEdgβ€˜π‘₯)(𝑓 β€œ ((iEdgβ€˜π‘₯)β€˜π‘–)) = ((iEdgβ€˜π‘¦)β€˜(π‘”β€˜π‘–)))})
 
Definitiondf-isomgr 45517* Define the isomorphy relation for graphs. (Contributed by AV, 11-Nov-2022.)
IsomGr = {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘“(𝑓:(Vtxβ€˜π‘₯)–1-1-ontoβ†’(Vtxβ€˜π‘¦) ∧ βˆƒπ‘”(𝑔:dom (iEdgβ€˜π‘₯)–1-1-ontoβ†’dom (iEdgβ€˜π‘¦) ∧ βˆ€π‘– ∈ dom (iEdgβ€˜π‘₯)(𝑓 β€œ ((iEdgβ€˜π‘₯)β€˜π‘–)) = ((iEdgβ€˜π‘¦)β€˜(π‘”β€˜π‘–))))}
 
Theoremisomgrrel 45518 The isomorphy relation for graphs is a relation. (Contributed by AV, 11-Nov-2022.)
Rel IsomGr
 
Theoremisomgr 45519* The isomorphy relation for two graphs. (Contributed by AV, 11-Nov-2022.)
𝑉 = (Vtxβ€˜π΄)    &   π‘Š = (Vtxβ€˜π΅)    &   πΌ = (iEdgβ€˜π΄)    &   π½ = (iEdgβ€˜π΅)    β‡’   ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ) β†’ (𝐴 IsomGr 𝐡 ↔ βˆƒπ‘“(𝑓:𝑉–1-1-ontoβ†’π‘Š ∧ βˆƒπ‘”(𝑔:dom 𝐼–1-1-ontoβ†’dom 𝐽 ∧ βˆ€π‘– ∈ dom 𝐼(𝑓 β€œ (πΌβ€˜π‘–)) = (π½β€˜(π‘”β€˜π‘–))))))
 
Theoremisisomgr 45520* Implications of two graphs being isomorphic. (Contributed by AV, 11-Nov-2022.)
𝑉 = (Vtxβ€˜π΄)    &   π‘Š = (Vtxβ€˜π΅)    &   πΌ = (iEdgβ€˜π΄)    &   π½ = (iEdgβ€˜π΅)    β‡’   (𝐴 IsomGr 𝐡 β†’ βˆƒπ‘“(𝑓:𝑉–1-1-ontoβ†’π‘Š ∧ βˆƒπ‘”(𝑔:dom 𝐼–1-1-ontoβ†’dom 𝐽 ∧ βˆ€π‘– ∈ dom 𝐼(𝑓 β€œ (πΌβ€˜π‘–)) = (π½β€˜(π‘”β€˜π‘–)))))
 
Theoremisomgreqve 45521 A set is isomorphic to a hypergraph if it has the same vertices and the same edges. (Contributed by AV, 11-Nov-2022.)
(((𝐴 ∈ UHGraph ∧ 𝐡 ∈ π‘Œ) ∧ ((Vtxβ€˜π΄) = (Vtxβ€˜π΅) ∧ (iEdgβ€˜π΄) = (iEdgβ€˜π΅))) β†’ 𝐴 IsomGr 𝐡)
 
Theoremisomushgr 45522* The isomorphy relation for two simple hypergraphs. (Contributed by AV, 28-Nov-2022.)
𝑉 = (Vtxβ€˜π΄)    &   π‘Š = (Vtxβ€˜π΅)    &   πΈ = (Edgβ€˜π΄)    &   πΎ = (Edgβ€˜π΅)    β‡’   ((𝐴 ∈ USHGraph ∧ 𝐡 ∈ USHGraph) β†’ (𝐴 IsomGr 𝐡 ↔ βˆƒπ‘“(𝑓:𝑉–1-1-ontoβ†’π‘Š ∧ βˆƒπ‘”(𝑔:𝐸–1-1-onto→𝐾 ∧ βˆ€π‘’ ∈ 𝐸 (𝑓 β€œ 𝑒) = (π‘”β€˜π‘’)))))
 
Theoremisomuspgrlem1 45523* Lemma 1 for isomuspgr 45530. (Contributed by AV, 29-Nov-2022.)
𝑉 = (Vtxβ€˜π΄)    &   π‘Š = (Vtxβ€˜π΅)    &   πΈ = (Edgβ€˜π΄)    &   πΎ = (Edgβ€˜π΅)    β‡’   (((((𝐴 ∈ USPGraph ∧ 𝐡 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-ontoβ†’π‘Š) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ βˆ€π‘’ ∈ 𝐸 (𝑓 β€œ 𝑒) = (π‘”β€˜π‘’))) ∧ (π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) β†’ ({(π‘“β€˜π‘Ž), (π‘“β€˜π‘)} ∈ 𝐾 β†’ {π‘Ž, 𝑏} ∈ 𝐸))
 
Theoremisomuspgrlem2a 45524* Lemma 1 for isomuspgrlem2 45529. (Contributed by AV, 29-Nov-2022.)
𝑉 = (Vtxβ€˜π΄)    &   π‘Š = (Vtxβ€˜π΅)    &   πΈ = (Edgβ€˜π΄)    &   πΎ = (Edgβ€˜π΅)    &   πΊ = (π‘₯ ∈ 𝐸 ↦ (𝐹 β€œ π‘₯))    β‡’   (𝐹 ∈ 𝑋 β†’ βˆ€π‘’ ∈ 𝐸 (𝐹 β€œ 𝑒) = (πΊβ€˜π‘’))
 
Theoremisomuspgrlem2b 45525* Lemma 2 for isomuspgrlem2 45529. (Contributed by AV, 29-Nov-2022.)
𝑉 = (Vtxβ€˜π΄)    &   π‘Š = (Vtxβ€˜π΅)    &   πΈ = (Edgβ€˜π΄)    &   πΎ = (Edgβ€˜π΅)    &   πΊ = (π‘₯ ∈ 𝐸 ↦ (𝐹 β€œ π‘₯))    &   (πœ‘ β†’ 𝐴 ∈ USPGraph)    &   (πœ‘ β†’ 𝐹:𝑉–1-1-ontoβ†’π‘Š)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝑉 βˆ€π‘ ∈ 𝑉 ({π‘Ž, 𝑏} ∈ 𝐸 ↔ {(πΉβ€˜π‘Ž), (πΉβ€˜π‘)} ∈ 𝐾))    β‡’   (πœ‘ β†’ 𝐺:𝐸⟢𝐾)
 
Theoremisomuspgrlem2c 45526* Lemma 3 for isomuspgrlem2 45529. (Contributed by AV, 29-Nov-2022.)
𝑉 = (Vtxβ€˜π΄)    &   π‘Š = (Vtxβ€˜π΅)    &   πΈ = (Edgβ€˜π΄)    &   πΎ = (Edgβ€˜π΅)    &   πΊ = (π‘₯ ∈ 𝐸 ↦ (𝐹 β€œ π‘₯))    &   (πœ‘ β†’ 𝐴 ∈ USPGraph)    &   (πœ‘ β†’ 𝐹:𝑉–1-1-ontoβ†’π‘Š)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝑉 βˆ€π‘ ∈ 𝑉 ({π‘Ž, 𝑏} ∈ 𝐸 ↔ {(πΉβ€˜π‘Ž), (πΉβ€˜π‘)} ∈ 𝐾))    &   (πœ‘ β†’ 𝐹 ∈ 𝑋)    β‡’   (πœ‘ β†’ 𝐺:𝐸–1-1→𝐾)
 
Theoremisomuspgrlem2d 45527* Lemma 4 for isomuspgrlem2 45529. (Contributed by AV, 1-Dec-2022.)
𝑉 = (Vtxβ€˜π΄)    &   π‘Š = (Vtxβ€˜π΅)    &   πΈ = (Edgβ€˜π΄)    &   πΎ = (Edgβ€˜π΅)    &   πΊ = (π‘₯ ∈ 𝐸 ↦ (𝐹 β€œ π‘₯))    &   (πœ‘ β†’ 𝐴 ∈ USPGraph)    &   (πœ‘ β†’ 𝐹:𝑉–1-1-ontoβ†’π‘Š)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝑉 βˆ€π‘ ∈ 𝑉 ({π‘Ž, 𝑏} ∈ 𝐸 ↔ {(πΉβ€˜π‘Ž), (πΉβ€˜π‘)} ∈ 𝐾))    &   (πœ‘ β†’ 𝐹 ∈ 𝑋)    &   (πœ‘ β†’ 𝐡 ∈ USPGraph)    β‡’   (πœ‘ β†’ 𝐺:𝐸–onto→𝐾)
 
Theoremisomuspgrlem2e 45528* Lemma 5 for isomuspgrlem2 45529. (Contributed by AV, 1-Dec-2022.)
𝑉 = (Vtxβ€˜π΄)    &   π‘Š = (Vtxβ€˜π΅)    &   πΈ = (Edgβ€˜π΄)    &   πΎ = (Edgβ€˜π΅)    &   πΊ = (π‘₯ ∈ 𝐸 ↦ (𝐹 β€œ π‘₯))    &   (πœ‘ β†’ 𝐴 ∈ USPGraph)    &   (πœ‘ β†’ 𝐹:𝑉–1-1-ontoβ†’π‘Š)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝑉 βˆ€π‘ ∈ 𝑉 ({π‘Ž, 𝑏} ∈ 𝐸 ↔ {(πΉβ€˜π‘Ž), (πΉβ€˜π‘)} ∈ 𝐾))    &   (πœ‘ β†’ 𝐹 ∈ 𝑋)    &   (πœ‘ β†’ 𝐡 ∈ USPGraph)    β‡’   (πœ‘ β†’ 𝐺:𝐸–1-1-onto→𝐾)
 
Theoremisomuspgrlem2 45529* Lemma 2 for isomuspgr 45530. (Contributed by AV, 1-Dec-2022.)
𝑉 = (Vtxβ€˜π΄)    &   π‘Š = (Vtxβ€˜π΅)    &   πΈ = (Edgβ€˜π΄)    &   πΎ = (Edgβ€˜π΅)    β‡’   (((𝐴 ∈ USPGraph ∧ 𝐡 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-ontoβ†’π‘Š) β†’ (βˆ€π‘Ž ∈ 𝑉 βˆ€π‘ ∈ 𝑉 ({π‘Ž, 𝑏} ∈ 𝐸 ↔ {(π‘“β€˜π‘Ž), (π‘“β€˜π‘)} ∈ 𝐾) β†’ βˆƒπ‘”(𝑔:𝐸–1-1-onto→𝐾 ∧ βˆ€π‘’ ∈ 𝐸 (𝑓 β€œ 𝑒) = (π‘”β€˜π‘’))))
 
Theoremisomuspgr 45530* The isomorphy relation for two simple pseudographs. This corresponds to the definition in [Bollobas] p. 3. (Contributed by AV, 1-Dec-2022.)
𝑉 = (Vtxβ€˜π΄)    &   π‘Š = (Vtxβ€˜π΅)    &   πΈ = (Edgβ€˜π΄)    &   πΎ = (Edgβ€˜π΅)    β‡’   ((𝐴 ∈ USPGraph ∧ 𝐡 ∈ USPGraph) β†’ (𝐴 IsomGr 𝐡 ↔ βˆƒπ‘“(𝑓:𝑉–1-1-ontoβ†’π‘Š ∧ βˆ€π‘Ž ∈ 𝑉 βˆ€π‘ ∈ 𝑉 ({π‘Ž, 𝑏} ∈ 𝐸 ↔ {(π‘“β€˜π‘Ž), (π‘“β€˜π‘)} ∈ 𝐾))))
 
Theoremisomgrref 45531 The isomorphy relation is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.)
(𝐺 ∈ UHGraph β†’ 𝐺 IsomGr 𝐺)
 
Theoremisomgrsym 45532 The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.)
((𝐴 ∈ UHGraph ∧ 𝐡 ∈ π‘Œ) β†’ (𝐴 IsomGr 𝐡 β†’ 𝐡 IsomGr 𝐴))
 
Theoremisomgrsymb 45533 The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.)
((𝐴 ∈ UHGraph ∧ 𝐡 ∈ UHGraph) β†’ (𝐴 IsomGr 𝐡 ↔ 𝐡 IsomGr 𝐴))
 
Theoremisomgrtrlem 45534* Lemma for isomgrtr 45535. (Contributed by AV, 5-Dec-2022.)
(((((𝐴 ∈ UHGraph ∧ 𝐡 ∈ UHGraph ∧ 𝐢 ∈ 𝑋) ∧ 𝑓:(Vtxβ€˜π΄)–1-1-ontoβ†’(Vtxβ€˜π΅) ∧ 𝑣:(Vtxβ€˜π΅)–1-1-ontoβ†’(Vtxβ€˜πΆ)) ∧ (𝑔:dom (iEdgβ€˜π΄)–1-1-ontoβ†’dom (iEdgβ€˜π΅) ∧ βˆ€π‘– ∈ dom (iEdgβ€˜π΄)(𝑓 β€œ ((iEdgβ€˜π΄)β€˜π‘–)) = ((iEdgβ€˜π΅)β€˜(π‘”β€˜π‘–)))) ∧ (𝑀:dom (iEdgβ€˜π΅)–1-1-ontoβ†’dom (iEdgβ€˜πΆ) ∧ βˆ€π‘˜ ∈ dom (iEdgβ€˜π΅)(𝑣 β€œ ((iEdgβ€˜π΅)β€˜π‘˜)) = ((iEdgβ€˜πΆ)β€˜(π‘€β€˜π‘˜)))) β†’ βˆ€π‘— ∈ dom (iEdgβ€˜π΄)((𝑣 ∘ 𝑓) β€œ ((iEdgβ€˜π΄)β€˜π‘—)) = ((iEdgβ€˜πΆ)β€˜((𝑀 ∘ 𝑔)β€˜π‘—)))
 
Theoremisomgrtr 45535 The isomorphy relation is transitive for hypergraphs. (Contributed by AV, 5-Dec-2022.)
((𝐴 ∈ UHGraph ∧ 𝐡 ∈ UHGraph ∧ 𝐢 ∈ 𝑋) β†’ ((𝐴 IsomGr 𝐡 ∧ 𝐡 IsomGr 𝐢) β†’ 𝐴 IsomGr 𝐢))
 
Theoremstrisomgrop 45536 A graph represented as an extensible structure with vertices as base set and indexed edges is isomorphic to a hypergraph represented as ordered pair with the same vertices and edges. (Contributed by AV, 11-Nov-2022.)
𝐺 = βŸ¨π‘‰, 𝐸⟩    &   π» = {⟨(Baseβ€˜ndx), π‘‰βŸ©, ⟨(.efβ€˜ndx), 𝐸⟩}    β‡’   ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ π‘Œ) β†’ 𝐺 IsomGr 𝐻)
 
Theoremushrisomgr 45537 A simple hypergraph (with arbitrarily indexed edges) is isomorphic to a graph with the same vertices and the same edges, indexed by the edges themselves. (Contributed by AV, 11-Nov-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΈ = (Edgβ€˜πΊ)    &   π» = βŸ¨π‘‰, ( I β†Ύ 𝐸)⟩    β‡’   (𝐺 ∈ USHGraph β†’ 𝐺 IsomGr 𝐻)
 
20.41.15.2  Loop-free graphs - extension
 
Theorem1hegrlfgr 45538* A graph 𝐺 with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.)
(πœ‘ β†’ 𝐴 ∈ 𝑋)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 β‰  𝐢)    &   (πœ‘ β†’ 𝐸 ∈ 𝒫 𝑉)    &   (πœ‘ β†’ (iEdgβ€˜πΊ) = {⟨𝐴, 𝐸⟩})    &   (πœ‘ β†’ {𝐡, 𝐢} βŠ† 𝐸)    β‡’   (πœ‘ β†’ (iEdgβ€˜πΊ):{𝐴}⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)})
 
20.41.15.3  Walks - extension
 
Syntaxcupwlks 45539 Extend class notation with walks (of a pseudograph).
class UPWalks
 
Definitiondf-upwlks 45540* Define the set of all walks (in a pseudograph), called "simple walks" in the following.

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)."

According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4.

Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n).

Although this definition is also applicable for arbitrary hypergraphs, it allows only walks consisting of not proper hyperedges (i.e. edges connecting at most two vertices). Therefore, it should be used for pseudographs only. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)

UPWalks = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom (iEdgβ€˜π‘”) ∧ 𝑝:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜π‘”) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})})
 
Theoremupwlksfval 45541* The set of simple walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 28-Dec-2020.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   (𝐺 ∈ π‘Š β†’ (UPWalksβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})})
 
Theoremisupwlk 45542* Properties of a pair of functions to be a simple walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   ((𝐺 ∈ π‘Š ∧ 𝐹 ∈ π‘ˆ ∧ 𝑃 ∈ 𝑍) β†’ (𝐹(UPWalksβ€˜πΊ)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})))
 
Theoremisupwlkg 45543* Generalization of isupwlk 45542: Conditions for two classes to represent a simple walk. (Contributed by AV, 5-Nov-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   (𝐺 ∈ π‘Š β†’ (𝐹(UPWalksβ€˜πΊ)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})))
 
Theoremupwlkbprop 45544 Basic properties of a simple walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 29-Dec-2020.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   (𝐹(UPWalksβ€˜πΊ)𝑃 β†’ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
 
Theoremupwlkwlk 45545 A simple walk is a walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 27-Feb-2021.)
(𝐹(UPWalksβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
 
Theoremupgrwlkupwlk 45546 In a pseudograph, a walk is a simple walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.)
((𝐺 ∈ UPGraph ∧ 𝐹(Walksβ€˜πΊ)𝑃) β†’ 𝐹(UPWalksβ€˜πΊ)𝑃)
 
Theoremupgrwlkupwlkb 45547 In a pseudograph, the definitions for a walk and a simple walk are equivalent. (Contributed by AV, 30-Dec-2020.)
(𝐺 ∈ UPGraph β†’ (𝐹(Walksβ€˜πΊ)𝑃 ↔ 𝐹(UPWalksβ€˜πΊ)𝑃))
 
TheoremupgrisupwlkALT 45548* Alternate proof of upgriswlk 28057 using the definition of UPGraph and related theorems. (Contributed by AV, 2-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ π‘ˆ ∧ 𝑃 ∈ 𝑍) β†’ (𝐹(Walksβ€˜πΊ)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})))
 
20.41.15.4  Edges of graphs expressed as sets of unordered pairs
 
Theoremupgredgssspr 45549 The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021.)
(𝐺 ∈ UPGraph β†’ (Edgβ€˜πΊ) βŠ† (Pairsβ€˜(Vtxβ€˜πΊ)))
 
Theoremuspgropssxp 45550* The set 𝐺 of "simple pseudographs" for a fixed set 𝑉 of vertices is a subset of a Cartesian product. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 45560. (Contributed by AV, 24-Nov-2021.)
𝑃 = 𝒫 (Pairsβ€˜π‘‰)    &   πΊ = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βˆƒπ‘ž ∈ USPGraph ((Vtxβ€˜π‘ž) = 𝑣 ∧ (Edgβ€˜π‘ž) = 𝑒))}    β‡’   (𝑉 ∈ π‘Š β†’ 𝐺 βŠ† (π‘Š Γ— 𝑃))
 
Theoremuspgrsprfv 45551* The value of the function 𝐹 which maps a "simple pseudograph" for a fixed set 𝑉 of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for 𝐺 as defined here, the function 𝐹 is a bijection between the "simple pseudographs" and the subsets of the set of pairs 𝑃 over the fixed set 𝑉 of vertices, see uspgrbispr 45557. (Contributed by AV, 24-Nov-2021.)
𝑃 = 𝒫 (Pairsβ€˜π‘‰)    &   πΊ = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βˆƒπ‘ž ∈ USPGraph ((Vtxβ€˜π‘ž) = 𝑣 ∧ (Edgβ€˜π‘ž) = 𝑒))}    &   πΉ = (𝑔 ∈ 𝐺 ↦ (2nd β€˜π‘”))    β‡’   (𝑋 ∈ 𝐺 β†’ (πΉβ€˜π‘‹) = (2nd β€˜π‘‹))
 
Theoremuspgrsprf 45552* The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 24-Nov-2021.)
𝑃 = 𝒫 (Pairsβ€˜π‘‰)    &   πΊ = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βˆƒπ‘ž ∈ USPGraph ((Vtxβ€˜π‘ž) = 𝑣 ∧ (Edgβ€˜π‘ž) = 𝑒))}    &   πΉ = (𝑔 ∈ 𝐺 ↦ (2nd β€˜π‘”))    β‡’   πΉ:πΊβŸΆπ‘ƒ
 
Theoremuspgrsprf1 45553* The mapping 𝐹 is a one-to-one function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.)
𝑃 = 𝒫 (Pairsβ€˜π‘‰)    &   πΊ = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βˆƒπ‘ž ∈ USPGraph ((Vtxβ€˜π‘ž) = 𝑣 ∧ (Edgβ€˜π‘ž) = 𝑒))}    &   πΉ = (𝑔 ∈ 𝐺 ↦ (2nd β€˜π‘”))    β‡’   πΉ:𝐺–1-1→𝑃
 
Theoremuspgrsprfo 45554* The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 onto the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.)
𝑃 = 𝒫 (Pairsβ€˜π‘‰)    &   πΊ = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βˆƒπ‘ž ∈ USPGraph ((Vtxβ€˜π‘ž) = 𝑣 ∧ (Edgβ€˜π‘ž) = 𝑒))}    &   πΉ = (𝑔 ∈ 𝐺 ↦ (2nd β€˜π‘”))    β‡’   (𝑉 ∈ π‘Š β†’ 𝐹:𝐺–onto→𝑃)
 
Theoremuspgrsprf1o 45555* The mapping 𝐹 is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. See also the comments on uspgrbisymrel 45560. (Contributed by AV, 25-Nov-2021.)
𝑃 = 𝒫 (Pairsβ€˜π‘‰)    &   πΊ = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βˆƒπ‘ž ∈ USPGraph ((Vtxβ€˜π‘ž) = 𝑣 ∧ (Edgβ€˜π‘ž) = 𝑒))}    &   πΉ = (𝑔 ∈ 𝐺 ↦ (2nd β€˜π‘”))    β‡’   (𝑉 ∈ π‘Š β†’ 𝐹:𝐺–1-1-onto→𝑃)
 
Theoremuspgrex 45556* The class 𝐺 of all "simple pseudographs" with a fixed set of vertices 𝑉 is a set. (Contributed by AV, 26-Nov-2021.)
𝑃 = 𝒫 (Pairsβ€˜π‘‰)    &   πΊ = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βˆƒπ‘ž ∈ USPGraph ((Vtxβ€˜π‘ž) = 𝑣 ∧ (Edgβ€˜π‘ž) = 𝑒))}    β‡’   (𝑉 ∈ π‘Š β†’ 𝐺 ∈ V)
 
Theoremuspgrbispr 45557* There is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 26-Nov-2021.)
𝑃 = 𝒫 (Pairsβ€˜π‘‰)    &   πΊ = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βˆƒπ‘ž ∈ USPGraph ((Vtxβ€˜π‘ž) = 𝑣 ∧ (Edgβ€˜π‘ž) = 𝑒))}    β‡’   (𝑉 ∈ π‘Š β†’ βˆƒπ‘“ 𝑓:𝐺–1-1-onto→𝑃)
 
Theoremuspgrspren 45558* The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑃 of subsets of the set of pairs over the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.)
𝑃 = 𝒫 (Pairsβ€˜π‘‰)    &   πΊ = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βˆƒπ‘ž ∈ USPGraph ((Vtxβ€˜π‘ž) = 𝑣 ∧ (Edgβ€˜π‘ž) = 𝑒))}    β‡’   (𝑉 ∈ π‘Š β†’ 𝐺 β‰ˆ 𝑃)
 
Theoremuspgrymrelen 45559* The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑅 of the symmetric relations on the fixed set 𝑉 are equinumerous. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 45560. (Contributed by AV, 27-Nov-2021.)
𝐺 = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βˆƒπ‘ž ∈ USPGraph ((Vtxβ€˜π‘ž) = 𝑣 ∧ (Edgβ€˜π‘ž) = 𝑒))}    &   π‘… = {π‘Ÿ ∈ 𝒫 (𝑉 Γ— 𝑉) ∣ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 (π‘₯π‘Ÿπ‘¦ ↔ π‘¦π‘Ÿπ‘₯)}    β‡’   (𝑉 ∈ π‘Š β†’ 𝐺 β‰ˆ 𝑅)
 
Theoremuspgrbisymrel 45560* There is a bijection between the "simple pseudographs" for a fixed set 𝑉 of vertices and the class 𝑅 of the symmetric relations on the fixed set 𝑉. The simple pseudographs, which are graphs without hyper- or multiedges, but which may contain loops, are expressed as ordered pairs of the vertices and the edges (as proper or improper unordered pairs of vertices, not as indexed edges!) in this theorem. That class 𝐺 of such simple pseudographs is a set (if 𝑉 is a set, see uspgrex 45556) of equivalence classes of graphs abstracting from the index sets of their edge functions.

Solely for this abstraction, there is a bijection between the "simple pseudographs" as members of 𝐺 and the symmetric relations 𝑅 on the fixed set 𝑉 of vertices. This theorem would not hold for 𝐺 = {𝑔 ∈ USPGraph ∣ (Vtxβ€˜π‘”) = 𝑉} and even not for 𝐺 = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βŸ¨π‘£, π‘’βŸ© ∈ USPGraph)}, because these are much bigger classes. (Proposed by Gerard Lang, 16-Nov-2021.) (Contributed by AV, 27-Nov-2021.)

𝐺 = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βˆƒπ‘ž ∈ USPGraph ((Vtxβ€˜π‘ž) = 𝑣 ∧ (Edgβ€˜π‘ž) = 𝑒))}    &   π‘… = {π‘Ÿ ∈ 𝒫 (𝑉 Γ— 𝑉) ∣ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 (π‘₯π‘Ÿπ‘¦ ↔ π‘¦π‘Ÿπ‘₯)}    β‡’   (𝑉 ∈ π‘Š β†’ βˆƒπ‘“ 𝑓:𝐺–1-1-onto→𝑅)
 
TheoremuspgrbisymrelALT 45561* Alternate proof of uspgrbisymrel 45560 not using the definition of equinumerosity. (Contributed by AV, 26-Nov-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 = {βŸ¨π‘£, π‘’βŸ© ∣ (𝑣 = 𝑉 ∧ βˆƒπ‘ž ∈ USPGraph ((Vtxβ€˜π‘ž) = 𝑣 ∧ (Edgβ€˜π‘ž) = 𝑒))}    &   π‘… = {π‘Ÿ ∈ 𝒫 (𝑉 Γ— 𝑉) ∣ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 (π‘₯π‘Ÿπ‘¦ ↔ π‘¦π‘Ÿπ‘₯)}    β‡’   (𝑉 ∈ π‘Š β†’ βˆƒπ‘“ 𝑓:𝐺–1-1-onto→𝑅)
 
20.41.16  Monoids (extension)
 
20.41.16.1  Auxiliary theorems
 
Theoremovn0dmfun 45562 If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6844. (Contributed by AV, 27-Jan-2020.)
((𝐴𝐹𝐡) β‰  βˆ… β†’ (⟨𝐴, 𝐡⟩ ∈ dom 𝐹 ∧ Fun (𝐹 β†Ύ {⟨𝐴, 𝐡⟩})))
 
Theoremxpsnopab 45563* A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.)
({𝑋} Γ— 𝐢) = {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž = 𝑋 ∧ 𝑏 ∈ 𝐢)}
 
Theoremxpiun 45564* A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.)
(𝐡 Γ— 𝐢) = βˆͺ π‘₯ ∈ 𝐡 {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž = π‘₯ ∧ 𝑏 ∈ 𝐢)}
 
Theoremovn0ssdmfun 45565* If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6844. (Contributed by AV, 27-Jan-2020.)
(βˆ€π‘Ž ∈ 𝐷 βˆ€π‘ ∈ 𝐸 (π‘ŽπΉπ‘) β‰  βˆ… β†’ ((𝐷 Γ— 𝐸) βŠ† dom 𝐹 ∧ Fun (𝐹 β†Ύ (𝐷 Γ— 𝐸))))
 
Theoremfnxpdmdm 45566 The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.)
(𝐹 Fn (𝐴 Γ— 𝐴) β†’ dom dom 𝐹 = 𝐴)
 
Theoremcnfldsrngbas 45567 The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
𝑅 = (β„‚fld β†Ύs 𝑆)    β‡’   (𝑆 βŠ† β„‚ β†’ 𝑆 = (Baseβ€˜π‘…))
 
Theoremcnfldsrngadd 45568 The group addition operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
𝑅 = (β„‚fld β†Ύs 𝑆)    β‡’   (𝑆 ∈ 𝑉 β†’ + = (+gβ€˜π‘…))
 
Theoremcnfldsrngmul 45569 The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
𝑅 = (β„‚fld β†Ύs 𝑆)    β‡’   (𝑆 ∈ 𝑉 β†’ Β· = (.rβ€˜π‘…))
 
20.41.16.2  Magmas, Semigroups and Monoids (extension)
 
Theoremplusfreseq 45570 If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily a magma), the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.)
𝐡 = (Baseβ€˜π‘€)    &    + = (+gβ€˜π‘€)    &    ⨣ = (+π‘“β€˜π‘€)    β‡’   (βˆ… βˆ‰ ran ⨣ β†’ ( + β†Ύ (𝐡 Γ— 𝐡)) = ⨣ )
 
Theoremmgmplusfreseq 45571 If the empty set is not contained in the base set of a magma, the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.)
𝐡 = (Baseβ€˜π‘€)    &    + = (+gβ€˜π‘€)    &    ⨣ = (+π‘“β€˜π‘€)    β‡’   ((𝑀 ∈ Mgm ∧ βˆ… βˆ‰ 𝐡) β†’ ( + β†Ύ (𝐡 Γ— 𝐡)) = ⨣ )
 
Theorem0mgm 45572 A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.)
(Baseβ€˜π‘€) = βˆ…    β‡’   (𝑀 ∈ 𝑉 β†’ 𝑀 ∈ Mgm)
 
Theoremmgmpropd 45573* If two structures have the same (nonempty) base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a magma iff the other one is. (Contributed by AV, 25-Feb-2020.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   (πœ‘ β†’ 𝐡 β‰  βˆ…)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))
 
Theoremismgmd 45574* Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΊ))    &   (πœ‘ β†’ 𝐺 ∈ 𝑉)    &   (πœ‘ β†’ + = (+gβ€˜πΊ))    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ + 𝑦) ∈ 𝐡)    β‡’   (πœ‘ β†’ 𝐺 ∈ Mgm)
 
20.41.16.3  Magma homomorphisms and submagmas
 
Syntaxcmgmhm 45575 Hom-set generator class for magmas.
class MgmHom
 
Syntaxcsubmgm 45576 Class function taking a magma to its lattice of submagmas.
class SubMgm
 
Definitiondf-mgmhm 45577* A magma homomorphism is a function on the base sets which preserves the binary operation. (Contributed by AV, 24-Feb-2020.)
MgmHom = (𝑠 ∈ Mgm, 𝑑 ∈ Mgm ↦ {𝑓 ∈ ((Baseβ€˜π‘‘) ↑m (Baseβ€˜π‘ )) ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘ )βˆ€π‘¦ ∈ (Baseβ€˜π‘ )(π‘“β€˜(π‘₯(+gβ€˜π‘ )𝑦)) = ((π‘“β€˜π‘₯)(+gβ€˜π‘‘)(π‘“β€˜π‘¦))})
 
Definitiondf-submgm 45578* A submagma is a subset of a magma which is closed under the operation. Such subsets are themselves magmas. (Contributed by AV, 24-Feb-2020.)
SubMgm = (𝑠 ∈ Mgm ↦ {𝑑 ∈ 𝒫 (Baseβ€˜π‘ ) ∣ βˆ€π‘₯ ∈ 𝑑 βˆ€π‘¦ ∈ 𝑑 (π‘₯(+gβ€˜π‘ )𝑦) ∈ 𝑑})
 
Theoremmgmhmrcl 45579 Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.)
(𝐹 ∈ (𝑆 MgmHom 𝑇) β†’ (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
 
Theoremsubmgmrcl 45580 Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.)
(𝑆 ∈ (SubMgmβ€˜π‘€) β†’ 𝑀 ∈ Mgm)
 
Theoremismgmhm 45581* Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
𝐡 = (Baseβ€˜π‘†)    &   πΆ = (Baseβ€˜π‘‡)    &    + = (+gβ€˜π‘†)    &    ⨣ = (+gβ€˜π‘‡)    β‡’   (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐡⟢𝐢 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ + 𝑦)) = ((πΉβ€˜π‘₯) ⨣ (πΉβ€˜π‘¦)))))
 
Theoremmgmhmf 45582 A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.)
𝐡 = (Baseβ€˜π‘†)    &   πΆ = (Baseβ€˜π‘‡)    β‡’   (𝐹 ∈ (𝑆 MgmHom 𝑇) β†’ 𝐹:𝐡⟢𝐢)
 
Theoremmgmhmpropd 45583* Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π½))    &   (πœ‘ β†’ 𝐢 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   (πœ‘ β†’ 𝐢 = (Baseβ€˜π‘€))    &   (πœ‘ β†’ 𝐡 β‰  βˆ…)    &   (πœ‘ β†’ 𝐢 β‰  βˆ…)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜π½)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜π‘€)𝑦))    β‡’   (πœ‘ β†’ (𝐽 MgmHom 𝐾) = (𝐿 MgmHom 𝑀))
 
Theoremmgmhmlin 45584 A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.)
𝐡 = (Baseβ€˜π‘†)    &    + = (+gβ€˜π‘†)    &    ⨣ = (+gβ€˜π‘‡)    β‡’   ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜(𝑋 + π‘Œ)) = ((πΉβ€˜π‘‹) ⨣ (πΉβ€˜π‘Œ)))
 
Theoremmgmhmf1o 45585 A magma homomorphism is bijective iff its converse is also a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 MgmHom 𝑆) β†’ (𝐹:𝐡–1-1-onto→𝐢 ↔ ◑𝐹 ∈ (𝑆 MgmHom 𝑅)))
 
Theoremidmgmhm 45586 The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.)
𝐡 = (Baseβ€˜π‘€)    β‡’   (𝑀 ∈ Mgm β†’ ( I β†Ύ 𝐡) ∈ (𝑀 MgmHom 𝑀))
 
Theoremissubmgm 45587* Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.)
𝐡 = (Baseβ€˜π‘€)    &    + = (+gβ€˜π‘€)    β‡’   (𝑀 ∈ Mgm β†’ (𝑆 ∈ (SubMgmβ€˜π‘€) ↔ (𝑆 βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑆 (π‘₯ + 𝑦) ∈ 𝑆)))
 
Theoremissubmgm2 45588 Submagmas are subsets that are also magmas. (Contributed by AV, 25-Feb-2020.)
𝐡 = (Baseβ€˜π‘€)    &   π» = (𝑀 β†Ύs 𝑆)    β‡’   (𝑀 ∈ Mgm β†’ (𝑆 ∈ (SubMgmβ€˜π‘€) ↔ (𝑆 βŠ† 𝐡 ∧ 𝐻 ∈ Mgm)))
 
Theoremrabsubmgmd 45589* Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.)
𝐡 = (Baseβ€˜π‘€)    &    + = (+gβ€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ Mgm)    &   ((πœ‘ ∧ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ (πœƒ ∧ 𝜏))) β†’ πœ‚)    &   (𝑧 = π‘₯ β†’ (πœ“ ↔ πœƒ))    &   (𝑧 = 𝑦 β†’ (πœ“ ↔ 𝜏))    &   (𝑧 = (π‘₯ + 𝑦) β†’ (πœ“ ↔ πœ‚))    β‡’   (πœ‘ β†’ {𝑧 ∈ 𝐡 ∣ πœ“} ∈ (SubMgmβ€˜π‘€))
 
Theoremsubmgmss 45590 Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.)
𝐡 = (Baseβ€˜π‘€)    β‡’   (𝑆 ∈ (SubMgmβ€˜π‘€) β†’ 𝑆 βŠ† 𝐡)
 
Theoremsubmgmid 45591 Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020.)
𝐡 = (Baseβ€˜π‘€)    β‡’   (𝑀 ∈ Mgm β†’ 𝐡 ∈ (SubMgmβ€˜π‘€))
 
Theoremsubmgmcl 45592 Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.)
+ = (+gβ€˜π‘€)    β‡’   ((𝑆 ∈ (SubMgmβ€˜π‘€) ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
 
Theoremsubmgmmgm 45593 Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020.)
𝐻 = (𝑀 β†Ύs 𝑆)    β‡’   (𝑆 ∈ (SubMgmβ€˜π‘€) β†’ 𝐻 ∈ Mgm)
 
Theoremsubmgmbas 45594 The base set of a submagma. (Contributed by AV, 26-Feb-2020.)
𝐻 = (𝑀 β†Ύs 𝑆)    β‡’   (𝑆 ∈ (SubMgmβ€˜π‘€) β†’ 𝑆 = (Baseβ€˜π»))
 
Theoremsubsubmgm 45595 A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.)
𝐻 = (𝐺 β†Ύs 𝑆)    β‡’   (𝑆 ∈ (SubMgmβ€˜πΊ) β†’ (𝐴 ∈ (SubMgmβ€˜π») ↔ (𝐴 ∈ (SubMgmβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)))
 
Theoremresmgmhm 45596 Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)
π‘ˆ = (𝑆 β†Ύs 𝑋)    β‡’   ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgmβ€˜π‘†)) β†’ (𝐹 β†Ύ 𝑋) ∈ (π‘ˆ MgmHom 𝑇))
 
Theoremresmgmhm2 45597 One direction of resmgmhm2b 45598. (Contributed by AV, 26-Feb-2020.)
π‘ˆ = (𝑇 β†Ύs 𝑋)    β‡’   ((𝐹 ∈ (𝑆 MgmHom π‘ˆ) ∧ 𝑋 ∈ (SubMgmβ€˜π‘‡)) β†’ 𝐹 ∈ (𝑆 MgmHom 𝑇))
 
Theoremresmgmhm2b 45598 Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.)
π‘ˆ = (𝑇 β†Ύs 𝑋)    β‡’   ((𝑋 ∈ (SubMgmβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom π‘ˆ)))
 
Theoremmgmhmco 45599 The composition of magma homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
((𝐹 ∈ (𝑇 MgmHom π‘ˆ) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) β†’ (𝐹 ∘ 𝐺) ∈ (𝑆 MgmHom π‘ˆ))
 
Theoremmgmhmima 45600 The homomorphic image of a submagma is a submagma. (Contributed by AV, 27-Feb-2020.)
((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgmβ€˜π‘€)) β†’ (𝐹 β€œ 𝑋) ∈ (SubMgmβ€˜π‘))
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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 46701-46765
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