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Type | Label | Description |
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Statement | ||
Theorem | bgoldbtbndlem1 45501 | Lemma 1 for bgoldbtbnd 45505: 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 45502* | Lemma 2 for bgoldbtbnd 45505. (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 45503* | Lemma 3 for bgoldbtbnd 45505. (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 45504* | Lemma 4 for bgoldbtbnd 45505. (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 45505* | 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 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.) |
β’ ((π β Even β§ 4 < π β§ π β€ (4 Β· (;10β;18))) β π β GoldbachEven ) | ||
Axiom | ax-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 (;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 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). 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 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). 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.) |
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 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.) |
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 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.) |
β’ ((π β Odd β§ 7 < π β§ π < (;88 Β· (;10β;29))) β π β GoldbachOdd ) | ||
Theorem | tgoldbachlt 45512* | 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 45511. (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 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.) |
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 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 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 45514 | Extend class notation to include the graph ispmorphisms. |
class GrIsom | ||
Syntax | cisomgr 45515 | Extend class notation to include the isomorphy relation for graphs. |
class IsomGr | ||
Definition | df-grisom 45516* | 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 45517* | 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 45518 | The isomorphy relation for graphs is a relation. (Contributed by AV, 11-Nov-2022.) |
β’ Rel IsomGr | ||
Theorem | isomgr 45519* | 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 45520* | 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 45521 | 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 45522* | 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 45523* | Lemma 1 for isomuspgr 45530. (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 45524* | Lemma 1 for isomuspgrlem2 45529. (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 45525* | Lemma 2 for isomuspgrlem2 45529. (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 45526* | Lemma 3 for isomuspgrlem2 45529. (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 45527* | Lemma 4 for isomuspgrlem2 45529. (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 45528* | Lemma 5 for isomuspgrlem2 45529. (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 45529* | Lemma 2 for isomuspgr 45530. (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 45530* | 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 45531 | The isomorphy relation is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.) |
β’ (πΊ β UHGraph β πΊ IsomGr πΊ) | ||
Theorem | isomgrsym 45532 | 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 45533 | 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 45534* | Lemma for isomgrtr 45535. (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 45535 | 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 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.) |
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 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.) |
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 45538* | 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 45539 | Extend class notation with walks (of a pseudograph). |
class UPWalks | ||
Definition | df-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.) |
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 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.) |
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 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.) |
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 45543* | Generalization of isupwlk 45542: 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 45544 | 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 45545 | 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 45546 | 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 45547 | 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 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.) |
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 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.) |
β’ (πΊ β UPGraph β (EdgβπΊ) β (Pairsβ(VtxβπΊ))) | ||
Theorem | uspgropssxp 45550* | 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 45560. (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 45551* | 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 45557. (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 45552* | 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 45553* | 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 45554* | 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 45555* | 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 45560. (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 45556* | 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 45557* | 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 45558* | 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 45559* | 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 45560. (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 45560* |
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 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 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 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.) |
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 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.) |
β’ ((π΄πΉπ΅) β β β (β¨π΄, π΅β© β dom πΉ β§ Fun (πΉ βΎ {β¨π΄, π΅β©}))) | ||
Theorem | xpsnopab 45563* | 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 45564* | 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 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.) |
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 45566 | 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 45567 | 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 45568 | 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 45569 | 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 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.) |
β’ π΅ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+πβπ) β β’ (β β ran ⨣ β ( + βΎ (π΅ Γ π΅)) = ⨣ ) | ||
Theorem | mgmplusfreseq 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.) |
β’ π΅ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+πβπ) β β’ ((π β Mgm β§ β β π΅) β ( + βΎ (π΅ Γ π΅)) = ⨣ ) | ||
Theorem | 0mgm 45572 | 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 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.) |
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 45574* | 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 45575 | Hom-set generator class for magmas. |
class MgmHom | ||
Syntax | csubmgm 45576 | Class function taking a magma to its lattice of submagmas. |
class SubMgm | ||
Definition | df-mgmhm 45577* | 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 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.) |
β’ SubMgm = (π β Mgm β¦ {π‘ β π« (Baseβπ ) β£ βπ₯ β π‘ βπ¦ β π‘ (π₯(+gβπ )π¦) β π‘}) | ||
Theorem | mgmhmrcl 45579 | 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 45580 | Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.) |
β’ (π β (SubMgmβπ) β π β Mgm) | ||
Theorem | ismgmhm 45581* | 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 45582 | 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 45583* | 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 45584 | 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 45585 | 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 45586 | 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)) | ||
Theorem | issubmgm 45587* | Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.) |
β’ π΅ = (Baseβπ) & β’ + = (+gβπ) β β’ (π β Mgm β (π β (SubMgmβπ) β (π β π΅ β§ βπ₯ β π βπ¦ β π (π₯ + π¦) β π))) | ||
Theorem | issubmgm2 45588 | Submagmas are subsets that are also magmas. (Contributed by AV, 25-Feb-2020.) |
β’ π΅ = (Baseβπ) & β’ π» = (π βΎs π) β β’ (π β Mgm β (π β (SubMgmβπ) β (π β π΅ β§ π» β Mgm))) | ||
Theorem | rabsubmgmd 45589* | Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.) |
β’ π΅ = (Baseβπ) & β’ + = (+gβπ) & β’ (π β π β Mgm) & β’ ((π β§ ((π₯ β π΅ β§ π¦ β π΅) β§ (π β§ π))) β π) & β’ (π§ = π₯ β (π β π)) & β’ (π§ = π¦ β (π β π)) & β’ (π§ = (π₯ + π¦) β (π β π)) β β’ (π β {π§ β π΅ β£ π} β (SubMgmβπ)) | ||
Theorem | submgmss 45590 | Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.) |
β’ π΅ = (Baseβπ) β β’ (π β (SubMgmβπ) β π β π΅) | ||
Theorem | submgmid 45591 | Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020.) |
β’ π΅ = (Baseβπ) β β’ (π β Mgm β π΅ β (SubMgmβπ)) | ||
Theorem | submgmcl 45592 | Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.) |
β’ + = (+gβπ) β β’ ((π β (SubMgmβπ) β§ π β π β§ π β π) β (π + π) β π) | ||
Theorem | submgmmgm 45593 | Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020.) |
β’ π» = (π βΎs π) β β’ (π β (SubMgmβπ) β π» β Mgm) | ||
Theorem | submgmbas 45594 | The base set of a submagma. (Contributed by AV, 26-Feb-2020.) |
β’ π» = (π βΎs π) β β’ (π β (SubMgmβπ) β π = (Baseβπ»)) | ||
Theorem | subsubmgm 45595 | A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.) |
β’ π» = (πΊ βΎs π) β β’ (π β (SubMgmβπΊ) β (π΄ β (SubMgmβπ») β (π΄ β (SubMgmβπΊ) β§ π΄ β π))) | ||
Theorem | resmgmhm 45596 | Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.) |
β’ π = (π βΎs π) β β’ ((πΉ β (π MgmHom π) β§ π β (SubMgmβπ)) β (πΉ βΎ π) β (π MgmHom π)) | ||
Theorem | resmgmhm2 45597 | One direction of resmgmhm2b 45598. (Contributed by AV, 26-Feb-2020.) |
β’ π = (π βΎs π) β β’ ((πΉ β (π MgmHom π) β§ π β (SubMgmβπ)) β πΉ β (π MgmHom π)) | ||
Theorem | resmgmhm2b 45598 | Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.) |
β’ π = (π βΎs π) β β’ ((π β (SubMgmβπ) β§ ran πΉ β π) β (πΉ β (π MgmHom π) β πΉ β (π MgmHom π))) | ||
Theorem | mgmhmco 45599 | The composition of magma homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.) |
β’ ((πΉ β (π MgmHom π) β§ πΊ β (π MgmHom π)) β (πΉ β πΊ) β (π MgmHom π)) | ||
Theorem | mgmhmima 45600 | The homomorphic image of a submagma is a submagma. (Contributed by AV, 27-Feb-2020.) |
β’ ((πΉ β (π MgmHom π) β§ π β (SubMgmβπ)) β (πΉ β π) β (SubMgmβπ)) |
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