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Type | Label | Description |
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Statement | ||
Theorem | nnsum4primesprm 45701* | Every prime is "the sum of at most 4" (actually one - the prime itself) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.) |
β’ (π β β β βπ β β βπ β (β βm (1...π))(π β€ 4 β§ π = Ξ£π β (1...π)(πβπ))) | ||
Theorem | nnsum3primesgbe 45702* | Any even Goldbach number is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.) |
β’ (π β GoldbachEven β βπ β β βπ β (β βm (1...π))(π β€ 3 β§ π = Ξ£π β (1...π)(πβπ))) | ||
Theorem | nnsum4primesgbe 45703* | Any even Goldbach number is the sum of at most 4 (actually 2) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.) |
β’ (π β GoldbachEven β βπ β β βπ β (β βm (1...π))(π β€ 4 β§ π = Ξ£π β (1...π)(πβπ))) | ||
Theorem | nnsum3primesle9 45704* | Every integer greater than 1 and less than or equal to 8 is the sum of at most 3 primes. (Contributed by AV, 2-Aug-2020.) |
β’ ((π β (β€β₯β2) β§ π β€ 8) β βπ β β βπ β (β βm (1...π))(π β€ 3 β§ π = Ξ£π β (1...π)(πβπ))) | ||
Theorem | nnsum4primesle9 45705* | Every integer greater than 1 and less than or equal to 8 is the sum of at most 4 primes. (Contributed by AV, 24-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.) |
β’ ((π β (β€β₯β2) β§ π β€ 8) β βπ β β βπ β (β βm (1...π))(π β€ 4 β§ π = Ξ£π β (1...π)(πβπ))) | ||
Theorem | nnsum4primesodd 45706* | If the (weak) ternary Goldbach conjecture is valid, then every odd integer greater than 5 is the sum of 3 primes. (Contributed by AV, 2-Jul-2020.) |
β’ (βπ β Odd (5 < π β π β GoldbachOddW ) β ((π β (β€β₯β6) β§ π β Odd ) β βπ β (β βm (1...3))π = Ξ£π β (1...3)(πβπ))) | ||
Theorem | nnsum4primesoddALTV 45707* | If the (strong) ternary Goldbach conjecture is valid, then every odd integer greater than 7 is the sum of 3 primes. (Contributed by AV, 26-Jul-2020.) |
β’ (βπ β Odd (7 < π β π β GoldbachOdd ) β ((π β (β€β₯β8) β§ π β Odd ) β βπ β (β βm (1...3))π = Ξ£π β (1...3)(πβπ))) | ||
Theorem | evengpop3 45708* | If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of an odd Goldbach number and 3. (Contributed by AV, 24-Jul-2020.) |
β’ (βπ β Odd (5 < π β π β GoldbachOddW ) β ((π β (β€β₯β9) β§ π β Even ) β βπ β GoldbachOddW π = (π + 3))) | ||
Theorem | evengpoap3 45709* | If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of an odd Goldbach number and 3. (Contributed by AV, 27-Jul-2020.) (Proof shortened by AV, 15-Sep-2021.) |
β’ (βπ β Odd (7 < π β π β GoldbachOdd ) β ((π β (β€β₯β;12) β§ π β Even ) β βπ β GoldbachOdd π = (π + 3))) | ||
Theorem | nnsum4primeseven 45710* | If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of 4 primes. (Contributed by AV, 25-Jul-2020.) |
β’ (βπ β Odd (5 < π β π β GoldbachOddW ) β ((π β (β€β₯β9) β§ π β Even ) β βπ β (β βm (1...4))π = Ξ£π β (1...4)(πβπ))) | ||
Theorem | nnsum4primesevenALTV 45711* | If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of 4 primes. (Contributed by AV, 27-Jul-2020.) |
β’ (βπ β Odd (7 < π β π β GoldbachOdd ) β ((π β (β€β₯β;12) β§ π β Even ) β βπ β (β βm (1...4))π = Ξ£π β (1...4)(πβπ))) | ||
Theorem | wtgoldbnnsum4prm 45712* | If the (weak) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes, showing that Schnirelmann's constant would be less than or equal to 4. See corollary 1.1 in [Helfgott] p. 4. (Contributed by AV, 25-Jul-2020.) |
β’ (βπ β Odd (5 < π β π β GoldbachOddW ) β βπ β (β€β₯β2)βπ β β βπ β (β βm (1...π))(π β€ 4 β§ π = Ξ£π β (1...π)(πβπ))) | ||
Theorem | stgoldbnnsum4prm 45713* | If the (strong) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes. (Contributed by AV, 27-Jul-2020.) |
β’ (βπ β Odd (7 < π β π β GoldbachOdd ) β βπ β (β€β₯β2)βπ β β βπ β (β βm (1...π))(π β€ 4 β§ π = Ξ£π β (1...π)(πβπ))) | ||
Theorem | bgoldbnnsum3prm 45714* | If the binary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 3 primes, showing that Schnirelmann's constant would be equal to 3. (Contributed by AV, 2-Aug-2020.) |
β’ (βπ β Even (4 < π β π β GoldbachEven ) β βπ β (β€β₯β2)βπ β β βπ β (β βm (1...π))(π β€ 3 β§ π = Ξ£π β (1...π)(πβπ))) | ||
Theorem | bgoldbtbndlem1 45715 | Lemma 1 for bgoldbtbnd 45719: the odd numbers between 7 and 13 (exclusive) are odd Goldbach numbers. (Contributed by AV, 29-Jul-2020.) |
β’ ((π β Odd β§ 7 < π β§ π β (7[,);13)) β π β GoldbachOdd ) | ||
Theorem | bgoldbtbndlem2 45716* | Lemma 2 for bgoldbtbnd 45719. (Contributed by AV, 1-Aug-2020.) |
β’ (π β π β (β€β₯β;11)) & β’ (π β π β (β€β₯β;11)) & β’ (π β βπ β Even ((4 < π β§ π < π) β π β GoldbachEven )) & β’ (π β π· β (β€β₯β3)) & β’ (π β πΉ β (RePartβπ·)) & β’ (π β βπ β (0..^π·)((πΉβπ) β (β β {2}) β§ ((πΉβ(π + 1)) β (πΉβπ)) < (π β 4) β§ 4 < ((πΉβ(π + 1)) β (πΉβπ)))) & β’ (π β (πΉβ0) = 7) & β’ (π β (πΉβ1) = ;13) & β’ (π β π < (πΉβπ·)) & β’ π = (π β (πΉβ(πΌ β 1))) β β’ ((π β§ π β Odd β§ πΌ β (1..^π·)) β ((π β ((πΉβπΌ)[,)(πΉβ(πΌ + 1))) β§ (π β (πΉβπΌ)) β€ 4) β (π β Even β§ π < π β§ 4 < π))) | ||
Theorem | bgoldbtbndlem3 45717* | Lemma 3 for bgoldbtbnd 45719. (Contributed by AV, 1-Aug-2020.) |
β’ (π β π β (β€β₯β;11)) & β’ (π β π β (β€β₯β;11)) & β’ (π β βπ β Even ((4 < π β§ π < π) β π β GoldbachEven )) & β’ (π β π· β (β€β₯β3)) & β’ (π β πΉ β (RePartβπ·)) & β’ (π β βπ β (0..^π·)((πΉβπ) β (β β {2}) β§ ((πΉβ(π + 1)) β (πΉβπ)) < (π β 4) β§ 4 < ((πΉβ(π + 1)) β (πΉβπ)))) & β’ (π β (πΉβ0) = 7) & β’ (π β (πΉβ1) = ;13) & β’ (π β π < (πΉβπ·)) & β’ (π β (πΉβπ·) β β) & β’ π = (π β (πΉβπΌ)) β β’ ((π β§ π β Odd β§ πΌ β (1..^π·)) β ((π β ((πΉβπΌ)[,)(πΉβ(πΌ + 1))) β§ 4 < π) β (π β Even β§ π < π β§ 4 < π))) | ||
Theorem | bgoldbtbndlem4 45718* | Lemma 4 for bgoldbtbnd 45719. (Contributed by AV, 1-Aug-2020.) |
β’ (π β π β (β€β₯β;11)) & β’ (π β π β (β€β₯β;11)) & β’ (π β βπ β Even ((4 < π β§ π < π) β π β GoldbachEven )) & β’ (π β π· β (β€β₯β3)) & β’ (π β πΉ β (RePartβπ·)) & β’ (π β βπ β (0..^π·)((πΉβπ) β (β β {2}) β§ ((πΉβ(π + 1)) β (πΉβπ)) < (π β 4) β§ 4 < ((πΉβ(π + 1)) β (πΉβπ)))) & β’ (π β (πΉβ0) = 7) & β’ (π β (πΉβ1) = ;13) & β’ (π β π < (πΉβπ·)) & β’ (π β (πΉβπ·) β β) β β’ (((π β§ πΌ β (1..^π·)) β§ π β Odd ) β ((π β ((πΉβπΌ)[,)(πΉβ(πΌ + 1))) β§ (π β (πΉβπΌ)) β€ 4) β βπ β β βπ β β βπ β β ((π β Odd β§ π β Odd β§ π β Odd ) β§ π = ((π + π) + π)))) | ||
Theorem | bgoldbtbnd 45719* | If the binary Goldbach conjecture is valid up to an integer π, and there is a series ("ladder") of primes with a difference of at most π up to an integer π, then the strong ternary Goldbach conjecture is valid up to π, 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.) |
β’ (π β π β (β€β₯β;11)) & β’ (π β π β (β€β₯β;11)) & β’ (π β βπ β Even ((4 < π β§ π < π) β π β GoldbachEven )) & β’ (π β π· β (β€β₯β3)) & β’ (π β πΉ β (RePartβπ·)) & β’ (π β βπ β (0..^π·)((πΉβπ) β (β β {2}) β§ ((πΉβ(π + 1)) β (πΉβπ)) < (π β 4) β§ 4 < ((πΉβ(π + 1)) β (πΉβπ)))) & β’ (π β (πΉβ0) = 7) & β’ (π β (πΉβ1) = ;13) & β’ (π β π < (πΉβπ·)) & β’ (π β (πΉβπ·) β β) β β’ (π β βπ β Odd ((7 < π β§ π < π) β π β GoldbachOdd )) | ||
Axiom | ax-bgbltosilva 45720 | 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.) |
β’ ((π β Even β§ 4 < π β§ π β€ (4 Β· (;10β;18))) β π β GoldbachEven ) | ||
Axiom | ax-tgoldbachgt 45721* | Temporary duplicate of tgoldbachgt 33037, provided as "axiom" as long as this theorem is in the mathbox of Thierry Arnoux: Odd integers greater than (;10β;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 πΊ of odd numbers which can be written as a sum of three odd primes. (Contributed by Thierry Arnoux, 22-Dec-2021.) |
β’ π = {π§ β β€ β£ Β¬ 2 β₯ π§} & β’ πΊ = {π§ β π β£ βπ β β βπ β β βπ β β ((π β π β§ π β π β§ π β π) β§ π§ = ((π + π) + π))} β β’ βπ β β (π β€ (;10β;27) β§ βπ β π (π < π β π β πΊ)) | ||
Theorem | tgoldbachgtALTV 45722* | Variant of Thierry Arnoux's tgoldbachgt 33037 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). This is proven by Helfgott (see section 7.4 in [Helfgott] p. 70) for π = 10^27. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 15-Jan-2022.) |
β’ βπ β β (π β€ (;10β;27) β§ βπ β Odd (π < π β π β GoldbachOdd )) | ||
Theorem | bgoldbachlt 45723* | 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). This is verified for m = 4 x 10^18 by Oliveira e Silva, see ax-bgbltosilva 45720. (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
β’ βπ β β ((4 Β· (;10β;18)) β€ π β§ βπ β Even ((4 < π β§ π < π) β π β GoldbachEven )) | ||
Axiom | ax-hgprmladder 45724 | 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.) |
β’ βπ β (β€β₯β3)βπ β (RePartβπ)(((πβ0) = 7 β§ (πβ1) = ;13 β§ (πβπ) = (;89 Β· (;10β;29))) β§ βπ β (0..^π)((πβπ) β (β β {2}) β§ ((πβ(π + 1)) β (πβπ)) < ((4 Β· (;10β;18)) β 4) β§ 4 < ((πβ(π + 1)) β (πβπ)))) | ||
Theorem | tgblthelfgott 45725 | 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 45723, ax-hgprmladder 45724 and bgoldbtbnd 45719. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
β’ ((π β Odd β§ 7 < π β§ π < (;88 Β· (;10β;29))) β π β GoldbachOdd ) | ||
Theorem | tgoldbachlt 45726* | The ternary Goldbach conjecture is valid for small odd numbers (i.e. for all odd numbers less than a fixed big π greater than 8 x 10^30). This is verified for m = 8.875694 x 10^30 by Helfgott, see tgblthelfgott 45725. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
β’ βπ β β ((8 Β· (;10β;30)) < π β§ βπ β Odd ((7 < π β§ π < π) β π β GoldbachOdd )) | ||
Theorem | tgoldbach 45727 | The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 45726 and ax-tgoldbachgt 45721. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
β’ βπ β Odd (7 < π β π β GoldbachOdd ) | ||
In the following, a general definition of the isomorphy relation for graphs and specializations for simple hypergraphs (isomushgr 45736) and simple pseudographs (isomuspgr 45744) 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 45745, isomgrsym 45746, isomgrtr 45749). Fianlly, isomorphic graphs with different representations are studied (strisomgrop 45750, ushrisomgr 45751). Maybe more important than graph isomorphy is the notion of graph isomorphism, which can be defined as in df-grisom 45730. Then π΄ IsomGr π΅ β βππ β (π΄ GrIsom π΅) resp. π΄ IsomGr π΅ β (π΄ GrIsom π΅) β β . Notice that there can be multiple isomorphisms between two graphs (let β¨{π΄, π΅}, {{π΄, π΅}}β© and β¨{{π, π}, {{π, π}}β© be two graphs with two vertices and one edge, then π΄ β¦ π, π΅ β¦ π and π΄ β¦ π, π΅ β¦ π 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. | ||
Syntax | cgrisom 45728 | Extend class notation to include the graph ispmorphisms. |
class GrIsom | ||
Syntax | cisomgr 45729 | Extend class notation to include the isomorphy relation for graphs. |
class IsomGr | ||
Definition | df-grisom 45730* | Define the class of all isomorphisms between two graphs. (Contributed by AV, 11-Dec-2022.) |
β’ GrIsom = (π₯ β V, π¦ β V β¦ {β¨π, πβ© β£ (π:(Vtxβπ₯)β1-1-ontoβ(Vtxβπ¦) β§ π:dom (iEdgβπ₯)β1-1-ontoβdom (iEdgβπ¦) β§ βπ β dom (iEdgβπ₯)(π β ((iEdgβπ₯)βπ)) = ((iEdgβπ¦)β(πβπ)))}) | ||
Definition | df-isomgr 45731* | Define the isomorphy relation for graphs. (Contributed by AV, 11-Nov-2022.) |
β’ IsomGr = {β¨π₯, π¦β© β£ βπ(π:(Vtxβπ₯)β1-1-ontoβ(Vtxβπ¦) β§ βπ(π:dom (iEdgβπ₯)β1-1-ontoβdom (iEdgβπ¦) β§ βπ β dom (iEdgβπ₯)(π β ((iEdgβπ₯)βπ)) = ((iEdgβπ¦)β(πβπ))))} | ||
Theorem | isomgrrel 45732 | The isomorphy relation for graphs is a relation. (Contributed by AV, 11-Nov-2022.) |
β’ Rel IsomGr | ||
Theorem | isomgr 45733* | The isomorphy relation for two graphs. (Contributed by AV, 11-Nov-2022.) |
β’ π = (Vtxβπ΄) & β’ π = (Vtxβπ΅) & β’ πΌ = (iEdgβπ΄) & β’ π½ = (iEdgβπ΅) β β’ ((π΄ β π β§ π΅ β π) β (π΄ IsomGr π΅ β βπ(π:πβ1-1-ontoβπ β§ βπ(π:dom πΌβ1-1-ontoβdom π½ β§ βπ β dom πΌ(π β (πΌβπ)) = (π½β(πβπ)))))) | ||
Theorem | isisomgr 45734* | Implications of two graphs being isomorphic. (Contributed by AV, 11-Nov-2022.) |
β’ π = (Vtxβπ΄) & β’ π = (Vtxβπ΅) & β’ πΌ = (iEdgβπ΄) & β’ π½ = (iEdgβπ΅) β β’ (π΄ IsomGr π΅ β βπ(π:πβ1-1-ontoβπ β§ βπ(π:dom πΌβ1-1-ontoβdom π½ β§ βπ β dom πΌ(π β (πΌβπ)) = (π½β(πβπ))))) | ||
Theorem | isomgreqve 45735 | A set is isomorphic to a hypergraph if it has the same vertices and the same edges. (Contributed by AV, 11-Nov-2022.) |
β’ (((π΄ β UHGraph β§ π΅ β π) β§ ((Vtxβπ΄) = (Vtxβπ΅) β§ (iEdgβπ΄) = (iEdgβπ΅))) β π΄ IsomGr π΅) | ||
Theorem | isomushgr 45736* | The isomorphy relation for two simple hypergraphs. (Contributed by AV, 28-Nov-2022.) |
β’ π = (Vtxβπ΄) & β’ π = (Vtxβπ΅) & β’ πΈ = (Edgβπ΄) & β’ πΎ = (Edgβπ΅) β β’ ((π΄ β USHGraph β§ π΅ β USHGraph) β (π΄ IsomGr π΅ β βπ(π:πβ1-1-ontoβπ β§ βπ(π:πΈβ1-1-ontoβπΎ β§ βπ β πΈ (π β π) = (πβπ))))) | ||
Theorem | isomuspgrlem1 45737* | Lemma 1 for isomuspgr 45744. (Contributed by AV, 29-Nov-2022.) |
β’ π = (Vtxβπ΄) & β’ π = (Vtxβπ΅) & β’ πΈ = (Edgβπ΄) & β’ πΎ = (Edgβπ΅) β β’ (((((π΄ β USPGraph β§ π΅ β USPGraph) β§ π:πβ1-1-ontoβπ) β§ (π:πΈβ1-1-ontoβπΎ β§ βπ β πΈ (π β π) = (πβπ))) β§ (π β π β§ π β π)) β ({(πβπ), (πβπ)} β πΎ β {π, π} β πΈ)) | ||
Theorem | isomuspgrlem2a 45738* | Lemma 1 for isomuspgrlem2 45743. (Contributed by AV, 29-Nov-2022.) |
β’ π = (Vtxβπ΄) & β’ π = (Vtxβπ΅) & β’ πΈ = (Edgβπ΄) & β’ πΎ = (Edgβπ΅) & β’ πΊ = (π₯ β πΈ β¦ (πΉ β π₯)) β β’ (πΉ β π β βπ β πΈ (πΉ β π) = (πΊβπ)) | ||
Theorem | isomuspgrlem2b 45739* | Lemma 2 for isomuspgrlem2 45743. (Contributed by AV, 29-Nov-2022.) |
β’ π = (Vtxβπ΄) & β’ π = (Vtxβπ΅) & β’ πΈ = (Edgβπ΄) & β’ πΎ = (Edgβπ΅) & β’ πΊ = (π₯ β πΈ β¦ (πΉ β π₯)) & β’ (π β π΄ β USPGraph) & β’ (π β πΉ:πβ1-1-ontoβπ) & β’ (π β βπ β π βπ β π ({π, π} β πΈ β {(πΉβπ), (πΉβπ)} β πΎ)) β β’ (π β πΊ:πΈβΆπΎ) | ||
Theorem | isomuspgrlem2c 45740* | Lemma 3 for isomuspgrlem2 45743. (Contributed by AV, 29-Nov-2022.) |
β’ π = (Vtxβπ΄) & β’ π = (Vtxβπ΅) & β’ πΈ = (Edgβπ΄) & β’ πΎ = (Edgβπ΅) & β’ πΊ = (π₯ β πΈ β¦ (πΉ β π₯)) & β’ (π β π΄ β USPGraph) & β’ (π β πΉ:πβ1-1-ontoβπ) & β’ (π β βπ β π βπ β π ({π, π} β πΈ β {(πΉβπ), (πΉβπ)} β πΎ)) & β’ (π β πΉ β π) β β’ (π β πΊ:πΈβ1-1βπΎ) | ||
Theorem | isomuspgrlem2d 45741* | Lemma 4 for isomuspgrlem2 45743. (Contributed by AV, 1-Dec-2022.) |
β’ π = (Vtxβπ΄) & β’ π = (Vtxβπ΅) & β’ πΈ = (Edgβπ΄) & β’ πΎ = (Edgβπ΅) & β’ πΊ = (π₯ β πΈ β¦ (πΉ β π₯)) & β’ (π β π΄ β USPGraph) & β’ (π β πΉ:πβ1-1-ontoβπ) & β’ (π β βπ β π βπ β π ({π, π} β πΈ β {(πΉβπ), (πΉβπ)} β πΎ)) & β’ (π β πΉ β π) & β’ (π β π΅ β USPGraph) β β’ (π β πΊ:πΈβontoβπΎ) | ||
Theorem | isomuspgrlem2e 45742* | Lemma 5 for isomuspgrlem2 45743. (Contributed by AV, 1-Dec-2022.) |
β’ π = (Vtxβπ΄) & β’ π = (Vtxβπ΅) & β’ πΈ = (Edgβπ΄) & β’ πΎ = (Edgβπ΅) & β’ πΊ = (π₯ β πΈ β¦ (πΉ β π₯)) & β’ (π β π΄ β USPGraph) & β’ (π β πΉ:πβ1-1-ontoβπ) & β’ (π β βπ β π βπ β π ({π, π} β πΈ β {(πΉβπ), (πΉβπ)} β πΎ)) & β’ (π β πΉ β π) & β’ (π β π΅ β USPGraph) β β’ (π β πΊ:πΈβ1-1-ontoβπΎ) | ||
Theorem | isomuspgrlem2 45743* | Lemma 2 for isomuspgr 45744. (Contributed by AV, 1-Dec-2022.) |
β’ π = (Vtxβπ΄) & β’ π = (Vtxβπ΅) & β’ πΈ = (Edgβπ΄) & β’ πΎ = (Edgβπ΅) β β’ (((π΄ β USPGraph β§ π΅ β USPGraph) β§ π:πβ1-1-ontoβπ) β (βπ β π βπ β π ({π, π} β πΈ β {(πβπ), (πβπ)} β πΎ) β βπ(π:πΈβ1-1-ontoβπΎ β§ βπ β πΈ (π β π) = (πβπ)))) | ||
Theorem | isomuspgr 45744* | The isomorphy relation for two simple pseudographs. This corresponds to the definition in [Bollobas] p. 3. (Contributed by AV, 1-Dec-2022.) |
β’ π = (Vtxβπ΄) & β’ π = (Vtxβπ΅) & β’ πΈ = (Edgβπ΄) & β’ πΎ = (Edgβπ΅) β β’ ((π΄ β USPGraph β§ π΅ β USPGraph) β (π΄ IsomGr π΅ β βπ(π:πβ1-1-ontoβπ β§ βπ β π βπ β π ({π, π} β πΈ β {(πβπ), (πβπ)} β πΎ)))) | ||
Theorem | isomgrref 45745 | The isomorphy relation is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.) |
β’ (πΊ β UHGraph β πΊ IsomGr πΊ) | ||
Theorem | isomgrsym 45746 | The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.) |
β’ ((π΄ β UHGraph β§ π΅ β π) β (π΄ IsomGr π΅ β π΅ IsomGr π΄)) | ||
Theorem | isomgrsymb 45747 | The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.) |
β’ ((π΄ β UHGraph β§ π΅ β UHGraph) β (π΄ IsomGr π΅ β π΅ IsomGr π΄)) | ||
Theorem | isomgrtrlem 45748* | Lemma for isomgrtr 45749. (Contributed by AV, 5-Dec-2022.) |
β’ (((((π΄ β UHGraph β§ π΅ β UHGraph β§ πΆ β π) β§ π:(Vtxβπ΄)β1-1-ontoβ(Vtxβπ΅) β§ π£:(Vtxβπ΅)β1-1-ontoβ(VtxβπΆ)) β§ (π:dom (iEdgβπ΄)β1-1-ontoβdom (iEdgβπ΅) β§ βπ β dom (iEdgβπ΄)(π β ((iEdgβπ΄)βπ)) = ((iEdgβπ΅)β(πβπ)))) β§ (π€:dom (iEdgβπ΅)β1-1-ontoβdom (iEdgβπΆ) β§ βπ β dom (iEdgβπ΅)(π£ β ((iEdgβπ΅)βπ)) = ((iEdgβπΆ)β(π€βπ)))) β βπ β dom (iEdgβπ΄)((π£ β π) β ((iEdgβπ΄)βπ)) = ((iEdgβπΆ)β((π€ β π)βπ))) | ||
Theorem | isomgrtr 45749 | The isomorphy relation is transitive for hypergraphs. (Contributed by AV, 5-Dec-2022.) |
β’ ((π΄ β UHGraph β§ π΅ β UHGraph β§ πΆ β π) β ((π΄ IsomGr π΅ β§ π΅ IsomGr πΆ) β π΄ IsomGr πΆ)) | ||
Theorem | strisomgrop 45750 | 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.) |
β’ πΊ = β¨π, πΈβ© & β’ π» = {β¨(Baseβndx), πβ©, β¨(.efβndx), πΈβ©} β β’ ((πΊ β UHGraph β§ π β π β§ πΈ β π) β πΊ IsomGr π») | ||
Theorem | ushrisomgr 45751 | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) & β’ π» = β¨π, ( I βΎ πΈ)β© β β’ (πΊ β USHGraph β πΊ IsomGr π») | ||
Theorem | 1hegrlfgr 45752* | A graph πΊ with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.) |
β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π΅ β πΆ) & β’ (π β πΈ β π« π) & β’ (π β (iEdgβπΊ) = {β¨π΄, πΈβ©}) & β’ (π β {π΅, πΆ} β πΈ) β β’ (π β (iEdgβπΊ):{π΄}βΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)}) | ||
Syntax | cupwlks 45753 | Extend class notation with walks (of a pseudograph). |
class UPWalks | ||
Definition | df-upwlks 45754* |
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.) |
β’ UPWalks = (π β V β¦ {β¨π, πβ© β£ (π β Word dom (iEdgβπ) β§ π:(0...(β―βπ))βΆ(Vtxβπ) β§ βπ β (0..^(β―βπ))((iEdgβπ)β(πβπ)) = {(πβπ), (πβ(π + 1))})}) | ||
Theorem | upwlksfval 45755* | The set of simple walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (πΊ β π β (UPWalksβπΊ) = {β¨π, πβ© β£ (π β Word dom πΌ β§ π:(0...(β―βπ))βΆπ β§ βπ β (0..^(β―βπ))(πΌβ(πβπ)) = {(πβπ), (πβ(π + 1))})}) | ||
Theorem | isupwlk 45756* | 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β π β§ πΉ β π β§ π β π) β (πΉ(UPWalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}))) | ||
Theorem | isupwlkg 45757* | Generalization of isupwlk 45756: Conditions for two classes to represent a simple walk. (Contributed by AV, 5-Nov-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (πΊ β π β (πΉ(UPWalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}))) | ||
Theorem | upwlkbprop 45758 | Basic properties of a simple walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 29-Dec-2020.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (πΉ(UPWalksβπΊ)π β (πΊ β V β§ πΉ β V β§ π β V)) | ||
Theorem | upwlkwlk 45759 | A simple walk is a walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 27-Feb-2021.) |
β’ (πΉ(UPWalksβπΊ)π β πΉ(WalksβπΊ)π) | ||
Theorem | upgrwlkupwlk 45760 | In a pseudograph, a walk is a simple walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.) |
β’ ((πΊ β UPGraph β§ πΉ(WalksβπΊ)π) β πΉ(UPWalksβπΊ)π) | ||
Theorem | upgrwlkupwlkb 45761 | In a pseudograph, the definitions for a walk and a simple walk are equivalent. (Contributed by AV, 30-Dec-2020.) |
β’ (πΊ β UPGraph β (πΉ(WalksβπΊ)π β πΉ(UPWalksβπΊ)π)) | ||
Theorem | upgrisupwlkALT 45762* | Alternate proof of upgriswlk 28375 using the definition of UPGraph and related theorems. (Contributed by AV, 2-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β UPGraph β§ πΉ β π β§ π β π) β (πΉ(WalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}))) | ||
Theorem | upgredgssspr 45763 | The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021.) |
β’ (πΊ β UPGraph β (EdgβπΊ) β (Pairsβ(VtxβπΊ))) | ||
Theorem | uspgropssxp 45764* | The set πΊ of "simple pseudographs" for a fixed set π of vertices is a subset of a Cartesian product. For more details about the class πΊ of all "simple pseudographs" see comments on uspgrbisymrel 45774. (Contributed by AV, 24-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} β β’ (π β π β πΊ β (π Γ π)) | ||
Theorem | uspgrsprfv 45765* | The value of the function πΉ which maps a "simple pseudograph" for a fixed set π of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for πΊ as defined here, the function πΉ is a bijection between the "simple pseudographs" and the subsets of the set of pairs π over the fixed set π of vertices, see uspgrbispr 45771. (Contributed by AV, 24-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ πΉ = (π β πΊ β¦ (2nd βπ)) β β’ (π β πΊ β (πΉβπ) = (2nd βπ)) | ||
Theorem | uspgrsprf 45766* | The mapping πΉ is a function from the "simple pseudographs" with a fixed set of vertices π into the subsets of the set of pairs over the set π. (Contributed by AV, 24-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ πΉ = (π β πΊ β¦ (2nd βπ)) β β’ πΉ:πΊβΆπ | ||
Theorem | uspgrsprf1 45767* | The mapping πΉ is a one-to-one function from the "simple pseudographs" with a fixed set of vertices π into the subsets of the set of pairs over the set π. (Contributed by AV, 25-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ πΉ = (π β πΊ β¦ (2nd βπ)) β β’ πΉ:πΊβ1-1βπ | ||
Theorem | uspgrsprfo 45768* | The mapping πΉ is a function from the "simple pseudographs" with a fixed set of vertices π onto the subsets of the set of pairs over the set π. (Contributed by AV, 25-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ πΉ = (π β πΊ β¦ (2nd βπ)) β β’ (π β π β πΉ:πΊβontoβπ) | ||
Theorem | uspgrsprf1o 45769* | The mapping πΉ is a bijection between the "simple pseudographs" with a fixed set of vertices π and the subsets of the set of pairs over the set π. See also the comments on uspgrbisymrel 45774. (Contributed by AV, 25-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ πΉ = (π β πΊ β¦ (2nd βπ)) β β’ (π β π β πΉ:πΊβ1-1-ontoβπ) | ||
Theorem | uspgrex 45770* | The class πΊ of all "simple pseudographs" with a fixed set of vertices π is a set. (Contributed by AV, 26-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} β β’ (π β π β πΊ β V) | ||
Theorem | uspgrbispr 45771* | There is a bijection between the "simple pseudographs" with a fixed set of vertices π and the subsets of the set of pairs over the set π. (Contributed by AV, 26-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} β β’ (π β π β βπ π:πΊβ1-1-ontoβπ) | ||
Theorem | uspgrspren 45772* | The set πΊ of the "simple pseudographs" with a fixed set of vertices π and the class π of subsets of the set of pairs over the fixed set π are equinumerous. (Contributed by AV, 27-Nov-2021.) |
β’ π = π« (Pairsβπ) & β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} β β’ (π β π β πΊ β π) | ||
Theorem | uspgrymrelen 45773* | The set πΊ of the "simple pseudographs" with a fixed set of vertices π and the class π of the symmetric relations on the fixed set π are equinumerous. For more details about the class πΊ of all "simple pseudographs" see comments on uspgrbisymrel 45774. (Contributed by AV, 27-Nov-2021.) |
β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ π = {π β π« (π Γ π) β£ βπ₯ β π βπ¦ β π (π₯ππ¦ β π¦ππ₯)} β β’ (π β π β πΊ β π ) | ||
Theorem | uspgrbisymrel 45774* |
There is a bijection between the "simple pseudographs" for a fixed
set
π of vertices and the class π
of the
symmetric relations on the
fixed set π. 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 πΊ of such simple pseudographs is a set
(if π is a
set, see uspgrex 45770) 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 πΊ and the symmetric relations π on the fixed set π of vertices. This theorem would not hold for πΊ = {π β USPGraph β£ (Vtxβπ) = π} and even not for πΊ = {β¨π£, πβ© β£ (π£ = π β§ β¨π£, πβ© β USPGraph)}, because these are much bigger classes. (Proposed by Gerard Lang, 16-Nov-2021.) (Contributed by AV, 27-Nov-2021.) |
β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ π = {π β π« (π Γ π) β£ βπ₯ β π βπ¦ β π (π₯ππ¦ β π¦ππ₯)} β β’ (π β π β βπ π:πΊβ1-1-ontoβπ ) | ||
Theorem | uspgrbisymrelALT 45775* | Alternate proof of uspgrbisymrel 45774 not using the definition of equinumerosity. (Contributed by AV, 26-Nov-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
β’ πΊ = {β¨π£, πβ© β£ (π£ = π β§ βπ β USPGraph ((Vtxβπ) = π£ β§ (Edgβπ) = π))} & β’ π = {π β π« (π Γ π) β£ βπ₯ β π βπ¦ β π (π₯ππ¦ β π¦ππ₯)} β β’ (π β π β βπ π:πΊβ1-1-ontoβπ ) | ||
Theorem | ovn0dmfun 45776 | 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 6881. (Contributed by AV, 27-Jan-2020.) |
β’ ((π΄πΉπ΅) β β β (β¨π΄, π΅β© β dom πΉ β§ Fun (πΉ βΎ {β¨π΄, π΅β©}))) | ||
Theorem | xpsnopab 45777* | A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.) |
β’ ({π} Γ πΆ) = {β¨π, πβ© β£ (π = π β§ π β πΆ)} | ||
Theorem | xpiun 45778* | A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.) |
β’ (π΅ Γ πΆ) = βͺ π₯ β π΅ {β¨π, πβ© β£ (π = π₯ β§ π β πΆ)} | ||
Theorem | ovn0ssdmfun 45779* | 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 6881. (Contributed by AV, 27-Jan-2020.) |
β’ (βπ β π· βπ β πΈ (ππΉπ) β β β ((π· Γ πΈ) β dom πΉ β§ Fun (πΉ βΎ (π· Γ πΈ)))) | ||
Theorem | fnxpdmdm 45780 | The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.) |
β’ (πΉ Fn (π΄ Γ π΄) β dom dom πΉ = π΄) | ||
Theorem | cnfldsrngbas 45781 | The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
β’ π = (βfld βΎs π) β β’ (π β β β π = (Baseβπ )) | ||
Theorem | cnfldsrngadd 45782 | The group addition operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
β’ π = (βfld βΎs π) β β’ (π β π β + = (+gβπ )) | ||
Theorem | cnfldsrngmul 45783 | The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
β’ π = (βfld βΎs π) β β’ (π β π β Β· = (.rβπ )) | ||
Theorem | plusfreseq 45784 | 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.) |
β’ π΅ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+πβπ) β β’ (β β ran ⨣ β ( + βΎ (π΅ Γ π΅)) = ⨣ ) | ||
Theorem | mgmplusfreseq 45785 | 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.) |
β’ π΅ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+πβπ) β β’ ((π β Mgm β§ β β π΅) β ( + βΎ (π΅ Γ π΅)) = ⨣ ) | ||
Theorem | 0mgm 45786 | A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.) |
β’ (Baseβπ) = β β β’ (π β π β π β Mgm) | ||
Theorem | mgmpropd 45787* | 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.) |
β’ (π β π΅ = (BaseβπΎ)) & β’ (π β π΅ = (BaseβπΏ)) & β’ (π β π΅ β β ) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) β β’ (π β (πΎ β Mgm β πΏ β Mgm)) | ||
Theorem | ismgmd 45788* | Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.) |
β’ (π β π΅ = (BaseβπΊ)) & β’ (π β πΊ β π) & β’ (π β + = (+gβπΊ)) & β’ ((π β§ π₯ β π΅ β§ π¦ β π΅) β (π₯ + π¦) β π΅) β β’ (π β πΊ β Mgm) | ||
Syntax | cmgmhm 45789 | Hom-set generator class for magmas. |
class MgmHom | ||
Syntax | csubmgm 45790 | Class function taking a magma to its lattice of submagmas. |
class SubMgm | ||
Definition | df-mgmhm 45791* | A magma homomorphism is a function on the base sets which preserves the binary operation. (Contributed by AV, 24-Feb-2020.) |
β’ MgmHom = (π β Mgm, π‘ β Mgm β¦ {π β ((Baseβπ‘) βm (Baseβπ )) β£ βπ₯ β (Baseβπ )βπ¦ β (Baseβπ )(πβ(π₯(+gβπ )π¦)) = ((πβπ₯)(+gβπ‘)(πβπ¦))}) | ||
Definition | df-submgm 45792* | 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.) |
β’ SubMgm = (π β Mgm β¦ {π‘ β π« (Baseβπ ) β£ βπ₯ β π‘ βπ¦ β π‘ (π₯(+gβπ )π¦) β π‘}) | ||
Theorem | mgmhmrcl 45793 | Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.) |
β’ (πΉ β (π MgmHom π) β (π β Mgm β§ π β Mgm)) | ||
Theorem | submgmrcl 45794 | Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.) |
β’ (π β (SubMgmβπ) β π β Mgm) | ||
Theorem | ismgmhm 45795* | Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.) |
β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+gβπ) β β’ (πΉ β (π MgmHom π) β ((π β Mgm β§ π β Mgm) β§ (πΉ:π΅βΆπΆ β§ βπ₯ β π΅ βπ¦ β π΅ (πΉβ(π₯ + π¦)) = ((πΉβπ₯) ⨣ (πΉβπ¦))))) | ||
Theorem | mgmhmf 45796 | A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.) |
β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) β β’ (πΉ β (π MgmHom π) β πΉ:π΅βΆπΆ) | ||
Theorem | mgmhmpropd 45797* | Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.) |
β’ (π β π΅ = (Baseβπ½)) & β’ (π β πΆ = (BaseβπΎ)) & β’ (π β π΅ = (BaseβπΏ)) & β’ (π β πΆ = (Baseβπ)) & β’ (π β π΅ β β ) & β’ (π β πΆ β β ) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπ½)π¦) = (π₯(+gβπΏ)π¦)) & β’ ((π β§ (π₯ β πΆ β§ π¦ β πΆ)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπ)π¦)) β β’ (π β (π½ MgmHom πΎ) = (πΏ MgmHom π)) | ||
Theorem | mgmhmlin 45798 | A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.) |
β’ π΅ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+gβπ) β β’ ((πΉ β (π MgmHom π) β§ π β π΅ β§ π β π΅) β (πΉβ(π + π)) = ((πΉβπ) ⨣ (πΉβπ))) | ||
Theorem | mgmhmf1o 45799 | A magma homomorphism is bijective iff its converse is also a magma homomorphism. (Contributed by AV, 25-Feb-2020.) |
β’ π΅ = (Baseβπ ) & β’ πΆ = (Baseβπ) β β’ (πΉ β (π MgmHom π) β (πΉ:π΅β1-1-ontoβπΆ β β‘πΉ β (π MgmHom π ))) | ||
Theorem | idmgmhm 45800 | The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.) |
β’ π΅ = (Baseβπ) β β’ (π β Mgm β ( I βΎ π΅) β (π MgmHom π)) |
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