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Type | Label | Description |
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Statement | ||
Theorem | bgoldbtbnd 45701* | 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 45702 | 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 45703* | 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 45704* | 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 45705* | 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 45702. (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 45706 | 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 45707 | 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 45705, ax-hgprmladder 45706 and bgoldbtbnd 45701. (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 45708* | 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 45707. (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 45709 | The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 45708 and ax-tgoldbachgt 45703. (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 45718) and simple pseudographs (isomuspgr 45726) 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 45727, isomgrsym 45728, isomgrtr 45731). Fianlly, isomorphic graphs with different representations are studied (strisomgrop 45732, ushrisomgr 45733). Maybe more important than graph isomorphy is the notion of graph isomorphism, which can be defined as in df-grisom 45712. 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 45710 | Extend class notation to include the graph ispmorphisms. |
class GrIsom | ||
Syntax | cisomgr 45711 | Extend class notation to include the isomorphy relation for graphs. |
class IsomGr | ||
Definition | df-grisom 45712* | 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 45713* | 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 45714 | The isomorphy relation for graphs is a relation. (Contributed by AV, 11-Nov-2022.) |
β’ Rel IsomGr | ||
Theorem | isomgr 45715* | 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 45716* | 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 45717 | 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 45718* | 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 45719* | Lemma 1 for isomuspgr 45726. (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 45720* | Lemma 1 for isomuspgrlem2 45725. (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 45721* | Lemma 2 for isomuspgrlem2 45725. (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 45722* | Lemma 3 for isomuspgrlem2 45725. (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 45723* | Lemma 4 for isomuspgrlem2 45725. (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 45724* | Lemma 5 for isomuspgrlem2 45725. (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 45725* | Lemma 2 for isomuspgr 45726. (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 45726* | 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 45727 | The isomorphy relation is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.) |
β’ (πΊ β UHGraph β πΊ IsomGr πΊ) | ||
Theorem | isomgrsym 45728 | 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 45729 | 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 45730* | Lemma for isomgrtr 45731. (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 45731 | 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 45732 | 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 45733 | 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 45734* | 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 45735 | Extend class notation with walks (of a pseudograph). |
class UPWalks | ||
Definition | df-upwlks 45736* |
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 45737* | 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 45738* | 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 45739* | Generalization of isupwlk 45738: 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 45740 | 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 45741 | 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 45742 | 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 45743 | 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 45744* | 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 45745 | 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 45746* | 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 45756. (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 45747* | 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 45753. (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 45748* | 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 45749* | 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 45750* | 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 45751* | 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 45756. (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 45752* | 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 45753* | 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 45754* | 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 45755* | 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 45756. (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 45756* |
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 45752) 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 45757* | Alternate proof of uspgrbisymrel 45756 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 45758 | 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 45759* | 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 45760* | 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 45761* | 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 45762 | 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 45763 | 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 45764 | 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 45765 | 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 45766 | 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 45767 | 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 45768 | 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 45769* | 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 45770* | 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 45771 | Hom-set generator class for magmas. |
class MgmHom | ||
Syntax | csubmgm 45772 | Class function taking a magma to its lattice of submagmas. |
class SubMgm | ||
Definition | df-mgmhm 45773* | 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 45774* | 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 45775 | 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 45776 | Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.) |
β’ (π β (SubMgmβπ) β π β Mgm) | ||
Theorem | ismgmhm 45777* | 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 45778 | 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 45779* | 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 45780 | 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 45781 | 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 45782 | 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 45783* | 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 45784 | 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 45785* | 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 45786 | 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 45787 | 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 45788 | 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 45789 | 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 45790 | 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 45791 | 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 45792 | 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 45793 | One direction of resmgmhm2b 45794. (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 45794 | 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 45795 | 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 45796 | 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)) | ||
Theorem | mgmhmeql 45797 | The equalizer of two magma homomorphisms is a submagma. (Contributed by AV, 27-Feb-2020.) |
β’ ((πΉ β (π MgmHom π) β§ πΊ β (π MgmHom π)) β dom (πΉ β© πΊ) β (SubMgmβπ)) | ||
Theorem | submgmacs 45798 | Submagmas are an algebraic closure system. (Contributed by AV, 27-Feb-2020.) |
β’ π΅ = (BaseβπΊ) β β’ (πΊ β Mgm β (SubMgmβπΊ) β (ACSβπ΅)) | ||
Theorem | ismhm0 45799 | Property of a monoid homomorphism, expressed by a magma homomorphism. (Contributed by AV, 17-Apr-2020.) |
β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+gβπ) & β’ 0 = (0gβπ) & β’ π = (0gβπ) β β’ (πΉ β (π MndHom π) β ((π β Mnd β§ π β Mnd) β§ (πΉ β (π MgmHom π) β§ (πΉβ 0 ) = π))) | ||
Theorem | mhmismgmhm 45800 | Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.) |
β’ (πΉ β (π MndHom π) β πΉ β (π MgmHom π)) |
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