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Theorem List for Metamath Proof Explorer - 21301-21400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremqrngdiv 21301 The division operation in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  Q  =  (flds  QQ )   =>    |-  ( ( X  e.  QQ  /\  Y  e.  QQ  /\  Y  =/=  0 ) 
 ->  ( X (/r `  Q ) Y )  =  ( X  /  Y ) )
 
Theoremqabvle 21302 By using induction on  N, we show a long-range inequality coming from the triangle inequality. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   =>    |-  ( ( F  e.  A  /\  N  e.  NN0 )  ->  ( F `  N )  <_  N )
 
Theoremqabvexp 21303 Induct the product rule abvmul 15900 to find the absolute value of a power. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   =>    |-  ( ( F  e.  A  /\  M  e.  QQ  /\  N  e.  NN0 )  ->  ( F `  ( M ^ N ) )  =  ( ( F `
  M ) ^ N ) )
 
Theoremostthlem1 21304* Lemma for ostth 21316. If two absolute values agree on the positive integers greater than one, then they agree for all rational numbers and thus are equal as functions. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ( ph  /\  n  e.  ( ZZ>= `  2 )
 )  ->  ( F `  n )  =  ( G `  n ) )   =>    |-  ( ph  ->  F  =  G )
 
Theoremostthlem2 21305* Lemma for ostth 21316. Refine ostthlem1 21304 so that it is sufficient to only show equality on the primes. (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ( ph  /\  p  e.  Prime )  ->  ( F `  p )  =  ( G `  p ) )   =>    |-  ( ph  ->  F  =  G )
 
Theoremqabsabv 21306 The regular absolute value function on the rationals is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   =>    |-  ( abs  |`  QQ )  e.  A
 
Theorempadicabv 21307* The p-adic absolute value (with arbitrary base) is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  F  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( N ^ ( P  pCnt  x ) ) ) )   =>    |-  ( ( P  e.  Prime  /\  N  e.  (
 0 (,) 1 ) ) 
 ->  F  e.  A )
 
Theorempadicabvf 21308* The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  J : Prime --> A
 
Theorempadicabvcxp 21309* All positive powers of the p-adic absolute value are absolute values. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   =>    |-  ( ( P  e.  Prime  /\  R  e.  RR+ )  ->  ( y  e. 
 QQ  |->  ( ( ( J `  P ) `
  y )  ^ c  R ) )  e.  A )
 
Theoremostth1 21310* - Lemma for ostth 21316: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If  F is equal to  1 on the primes, then by complete induction and the multiplicative property abvmul 15900 of the absolute value,  F is equal to  1 on all the integers, and ostthlem1 21304 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  A. n  e.  NN  -.  1  <  ( F `
  n ) )   &    |-  ( ph  ->  A. n  e. 
 Prime  -.  ( F `  n )  <  1 )   =>    |-  ( ph  ->  F  =  K )
 
Theoremostth2lem2 21311* Lemma for ostth2 21314. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  S  =  ( ( log `  ( F `  M ) ) 
 /  ( log `  M ) )   &    |-  T  =  if ( ( F `  M )  <_  1 ,  1 ,  ( F `
  M ) )   =>    |-  ( ( ph  /\  X  e.  NN0  /\  Y  e.  ( 0 ... (
 ( M ^ X )  -  1 ) ) )  ->  ( F `  Y )  <_  (
 ( M  x.  X )  x.  ( T ^ X ) ) )
 
Theoremostth2lem3 21312* Lemma for ostth2 21314. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  S  =  ( ( log `  ( F `  M ) ) 
 /  ( log `  M ) )   &    |-  T  =  if ( ( F `  M )  <_  1 ,  1 ,  ( F `
  M ) )   &    |-  U  =  ( ( log `  N )  /  ( log `  M )
 )   =>    |-  ( ( ph  /\  X  e.  NN )  ->  (
 ( ( F `  N )  /  ( T  ^ c  U ) ) ^ X ) 
 <_  ( X  x.  (
 ( M  x.  T )  x.  ( U  +  1 ) ) ) )
 
Theoremostth2lem4 21313* Lemma for ostth2 21314. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  2
 ) )   &    |-  S  =  ( ( log `  ( F `  M ) ) 
 /  ( log `  M ) )   &    |-  T  =  if ( ( F `  M )  <_  1 ,  1 ,  ( F `
  M ) )   &    |-  U  =  ( ( log `  N )  /  ( log `  M )
 )   =>    |-  ( ph  ->  (
 1  <  ( F `  M )  /\  R  <_  S ) )
 
Theoremostth2 21314* - Lemma for ostth 21316: regular case. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  1  <  ( F `  N ) )   &    |-  R  =  ( ( log `  ( F `  N ) ) 
 /  ( log `  N ) )   =>    |-  ( ph  ->  E. a  e.  ( 0 (,] 1
 ) F  =  ( y  e.  QQ  |->  ( ( abs `  y
 )  ^ c  a ) ) )
 
Theoremostth3 21315* - Lemma for ostth 21316: p-adic case. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  A. n  e.  NN  -.  1  <  ( F `
  n ) )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( F `  P )  <  1 )   &    |-  R  =  -u ( ( log `  ( F `  P ) )  /  ( log `  P ) )   &    |-  S  =  if (
 ( F `  P )  <_  ( F `  p ) ,  ( F `  p ) ,  ( F `  P ) )   =>    |-  ( ph  ->  E. a  e.  RR+  F  =  ( y  e.  QQ  |->  ( ( ( J `  P ) `  y
 )  ^ c  a ) ) )
 
Theoremostth 21316* Ostrowski's theorem, which classifies all absolute values on  QQ. Any such absolute value must either be the trivial absolute value  K, a constant exponent  0  <  a  <_  1 times the regular absolute value, or a positive exponent times the p-adic absolute value. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  Q  =  (flds  QQ )   &    |-  A  =  (AbsVal `  Q )   &    |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 , 
 0 ,  ( q ^ -u ( q  pCnt  x ) ) ) ) )   &    |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )   =>    |-  ( F  e.  A  <->  ( F  =  K  \/  E. a  e.  ( 0 (,] 1 ) F  =  ( y  e. 
 QQ  |->  ( ( abs `  y )  ^ c  a ) )  \/ 
 E. a  e.  RR+  E. g  e.  ran  J  F  =  ( y  e.  QQ  |->  ( ( g `
  y )  ^ c  a ) ) ) )
 
PART 14  GRAPH THEORY



To give an overview of the definitions and terms used in the context of graph theory, a glossary is provided in the following, mainly according to Definitions in [Bollobas] p. 1-8. Although this glossary concentrates on undirected graphs, many of the concepts are also useful for directed graphs.

Basic kinds of graphs:

TermReferenceDefinitionRemarks
(Undirected) Hypergraph df-uhgra 21318 an ordered pair  <. V ,  E >. of a set  V and a function  E into the powerset of  V ( ran  E  C_  ( ~P V )).
An element of  V is called "vertex", an element of  ran  E is called "edge", the function  E is called the "edge-function" .
In this most general definition of a graph, an "edge" may connect three or more vertices with each other, compare with the definition in Section I.1 in [Bollobas] p. 7.
Undirected multigraph df-umgra 21331 a graph  <. V ,  E >. such that  E is a function into the set of (proper or not proper) unordered pairs of  V.A proper unordered pair contains two different elements, a not proper unordered pair contains two times the same element, so it is a singleton (see preqsn 3967).
According to the definition in Section I.1 in [Bollobas] p. 7, "In a multigraph both multiple edges [joining two vertices] and multiple loops [joining a vertex to itself] are allowed".
Undirected simple graph with loops df-uslgra 21349 a graph  <. V ,  E >. such that  E is a one-to-one function into the set of (proper or not proper) unordered pairs of  V.This means that there is at most one edge between two vertices, and at most one loop from a vertex to itself.
Undirected simple graph without loops (in short "simple graph") df-usgra 21350 a graph  <. V ,  E >. such that  E is a one-to-one function into the set of (proper) unordered pairs of  V.An ordered pair  <. V ,  E >. of two distinct sets  V and  E (the "usual" definition of a "graph", see, for example, the definition in Section I.1 in [Bollobas] p. 1) can be identified with an undirected simple graph without loops by "indexing" the edges with themselves, see ausisusgra 21363.
Finite graph---a graph  <. V ,  E >. with finite sets  V and  E.In simple graphs,  E is finite if  V is finite, see usgrafis 21412. The number of edges is limited by  ( n  _C  2 ) (or " n choose 2") with  n  =  ( # `  V ), see usgramaxsize 21479. Analogously, the number of edges of an undirected simple graph with loops is limited by  ( ( n  +  1 )  _C  2 ). In multigraphs, however,  E can be infinite although  V is finite.
Graph of finite size---a graph  <. V ,  E >. with finite set  E, i.e. with a finite number of edges.A graph can be of finite size although  V is infinite.


Terms and properties of graphs:
TermReferenceDefinitionRemarks
Edge joining (two) vertices --- An edge  e  e.  ran  E "joins" the vertices v1, v2, ... vn ( n  e.  NN) if  e = { v1, v2, ... vn }. If  n  =  1,  e = { v1 } is a "loop", if  n  =  2,  e = { v1 , v2 } is an egde as it is usually defined, see definition in Section I.1 in [Bollobas] p. 1.
(Two) Endvertices of an edge see definition in Section I.1 in [Bollobas] p. 1. If an edge  e  e.  ran  E joins the vertices v1, v2, ... vn ( n  e.  NN), then the vertices v1, v2, ... vn are called the "endvertices" of the edge  e.
(Two) Adjacent vertices see definition in Section I.1 in [Bollobas] p. 1/2. The vertices v1, v2, ... vn ( n  e.  NN) are "adjacent" if there is an edge e = { v1, v2, ... vn } joining these vertices. In this case, the vertices are "incident" with the edge e (see definition in Section I.1 in [Bollobas] p. 2) or "connected" by the edge e.
(Two) Adjacent edges The edges e0, e1, ... en ( n  e.  NN) are "adjacent" if they have exactly one common endvertex. Generalization of definition in Section I.1 in [Bollobas] p. 2.
Order of a graph see definition in Section I.1 in [Bollobas] p. 3 the "order" of a graph  <. V ,  E >. is the number of vertices in the graph ( ( # `  V )).
Size of a graph see definition in Section I.1 in [Bollobas] p. 3 the "size" of a graph  <. V ,  E >. is the number of edges in the graph ( ( # `  E )).
Neighborhood of a vertex df-nbgra 21416 resp. definition in Section I.1 in [Bollobas] p. 3 A vertex connected with a vertex  v by an edge is called a "neighbor" of the vertex  v. The set of neighbors of a vertex  v is called the "neighborhood" (or "open neighborhood") of the vertex  v. The "closed neighborhood" is the union of the (open) neighborhood of the vertex  v with  { v }.
Degree of a vertex df-vdgr 21648 The "degree" of a vertex is the number of the edges having this vertex as endvertex. In a simple graph, the degree of a vertex is the number of neighbors of this vertex, see definition in Section I.1 in [Bollobas] p. 3
Isolated vertex usgravd0nedg 21666 A vertex is called "isolated" if it is not an endvertex of any edge, thus having degree 0.
Universal vertex df-uvtx 21418 A vertex is called "universal" if it is connected with every other vertex of the graph by an edge, thus having degree  ( # `  V ).


Special kinds of graphs:
TermReferenceDefinitionRemarks
Complete graph df-cusgra 21417 A graph is called "complete" if each pair of vertices is connected by an edge. The size of a complete undirected simple graph of order  n is  ( n  _C  2 ) (or " n choose 2"), see cusgrasize 21470.
Empty graph umgra0 21343 and usgra0 21373 A graph is called "empty" if it has no edges.
Null graph usgra0v 21374 A graph is called the "null graph" if it has no vertices (and therefore also no edges).
Trivial graph usgra1v 21392 A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph df-conngra 21640 resp. definition in Section I.1 in [Bollobas] p. 6 A graph is called "connected" if for each pair of vertices there is a path between these vertices.


For the terms "Path", "Walk", "Trail", "Circuit", "Cycle" see the remarks below and the definitions in Section I.1 in [Bollobas] p. 4-5.
 
14.1  Undirected graphs - basics
 
14.1.1  Undirected hypergraphs
 
Syntaxcuhg 21317 Extend class notation with undirected hypergraphs.
 class UHGrph
 
Definitiondf-uhgra 21318* Define the class of all undirected hypergraphs. An undirected hypergraph is a pair of a set and a function into the powerset of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |- UHGrph  =  { <. v ,  e >.  |  e : dom  e
 --> ( ~P v  \  { (/) } ) }
 
Theoremreluhgra 21319 The class of all undirected hypergraphs is a relation. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |- 
 Rel UHGrph
 
Theoremuhgrav 21320 The classes of vertices and edges of an undirected hypergraph are sets. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( V UHGrph  E  ->  ( V  e.  _V  /\  E  e.  _V )
 )
 
Theoremisuhgra 21321 The property of being an undirected hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V UHGrph  E  <->  E : dom  E --> ( ~P V  \  { (/) } )
 ) )
 
Theoremuhgraf 21322 The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( V UHGrph  E  ->  E : dom  E --> ( ~P V  \  { (/) } )
 )
 
Theoremuhgrafun 21323 The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( V UHGrph  E  ->  Fun 
 E )
 
Theoremuhgrass 21324 An edge is a subset of vertices, analogous to umgrass 21337. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( ( V UHGrph  E  /\  F  e.  dom  E )  ->  ( E `  F )  C_  V )
 
Theoremuhgraeq12d 21325 Equality of hypergraphs. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( V  =  W  /\  E  =  F )
 )  ->  ( V UHGrph  E  <->  W UHGrph  F ) )
 
Theoremuhgrares 21326 A subgraph of a hypergraph (formed by removing some edges from the original graph) is a hypergraph, analogous to umgrares 21342. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( V UHGrph  E  ->  V UHGrph 
 ( E  |`  A ) )
 
Theoremuhgra0 21327 The empty graph, with vertices but no edges, is a hypergraph, analogous to umgra0 21343. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( V  e.  W  ->  V UHGrph  (/) )
 
Theoremuhgra0v 21328 The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( (/) UHGrph  E  <->  E  =  (/) )
 
Theoremuhgraun 21329 If  <. V ,  E >. and  <. V ,  F >. are hypergraphs, then  <. V ,  E  u.  F >. is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting in two edges between two vertices), analogous to umgraun 21346. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UHGrph  E )   &    |-  ( ph  ->  V UHGrph  F )   =>    |-  ( ph  ->  V UHGrph 
 ( E  u.  F ) )
 
14.1.2  Undirected multigraphs
 
Syntaxcumg 21330 Extend class notation with undirected multigraphs.
 class UMGrph
 
Definitiondf-umgra 21331* Define the class of all undirected multigraphs. A multigraph is a pair  <. V ,  E >. where  E is a function into subsets of  V of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- UMGrph  =  { <. v ,  e >.  |  e : dom  e
 --> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
 
Theoremrelumgra 21332 The class of all undirected multigraphs is a relation. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- 
 Rel UMGrph
 
Theoremisumgra 21333* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremwrdumgra 21334* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V  e.  W  /\  E  e. Word  X )  ->  ( V UMGrph  E  <->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremumgraf2 21335* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremumgraf 21336* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A ) 
 ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremumgrass 21337 An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  C_  V )
 
Theoremumgran0 21338 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  =/=  (/) )
 
Theoremumgrale 21339 An edge has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( # `  ( E `  F ) ) 
 <_  2 )
 
Theoremumgrafi 21340 An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  e.  Fin )
 
Theoremumgraex 21341* An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  E. x  e.  V  E. y  e.  V  ( E `  F )  =  { x ,  y } )
 
Theoremumgrares 21342 A subgraph of a graph (formed by removing some edges from the original graph) is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V UMGrph  E  ->  V UMGrph 
 ( E  |`  A ) )
 
Theoremumgra0 21343 The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V  e.  W  ->  V UMGrph  (/) )
 
Theoremumgra1 21344 The graph with one edge. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V )
 )  ->  V UMGrph  { <. A ,  { B ,  C } >. } )
 
Theoremumisuhgra 21345 An undirected multigraph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( V UMGrph  E  ->  V UHGrph  E )
 
Theoremumgraun 21346 If  <. V ,  E >. and  <. V ,  F >. are graphs, then  <. V ,  E  u.  F >. is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UMGrph  E )   &    |-  ( ph  ->  V UMGrph  F )   =>    |-  ( ph  ->  V UMGrph 
 ( E  u.  F ) )
 
14.1.3  Undirected simple graphs
 
14.1.3.1  Undirected simple graphs - basics
 
Syntaxcuslg 21347 Extend class notation with undirected (simple) graphs with loops.
 class USLGrph
 
Syntaxcusg 21348 Extend class notation with undirected (simple) graphs (without loops).
 class USGrph
 
Definitiondf-uslgra 21349* Define the class of all undirected simple graphs with loops. An undirected simple graph with loops is a special undirected multigraph  <. V ,  E >. where  E is an injective (one-to-one) function into subsets of  V of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a multigraph, there is at most one edge between two vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- USLGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
 
Definitiondf-usgra 21350* Define the class of all undirected simple graphs without loops. An undirected simple graph without loops is a special undirected simple graph  <. V ,  E >. where 
E is an injective (one-to-one) function into subsets of  V of cardinality two, representing the two vertices incident to the edge. Such graphs are usually simply called "undirected graphs", so if only the term "undirected graph" is used, an undirected simple graph without loops is meant. Therefore, an undirected graph has no loops (edges a vertex to itsself). (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- USGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
 
Theoremreluslgra 21351 The class of all undirected simple graph with loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- 
 Rel USLGrph
 
Theoremrelusgra 21352 The class of all undirected simple graph without loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- 
 Rel USGrph
 
Theoremuslgrav 21353 The classes of vertices and edges of an undirected simple graph with loops are sets. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
 |-  ( V USLGrph  E  ->  ( V  e.  _V  /\  E  e.  _V )
 )
 
Theoremusgrav 21354 The classes of vertices and edges of an undirected simple graph without loops are sets. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  ( V USGrph  E  ->  ( V  e.  _V  /\  E  e.  _V )
 )
 
Theoremisuslgra 21355* The property of being an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USLGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremisusgra 21356* The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
 ) )
 
Theoremuslgraf 21357* The edge function of an undirected simple graph with loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USLGrph  E  ->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremusgraf 21358* The edge function of an undirected simple graph without loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
 )
 
Theoremisusgra0 21359* The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }
 ) )
 
Theoremusgraf0 21360* The edge function of an undirected simple graph without loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }
 )
 
Theoremusgrafun 21361 The edge function of an undirected simple graph without loops is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017.)
 |-  ( V USGrph  E  ->  Fun 
 E )
 
Theoremisausgra 21362* The property of an unordered pair to be an alternatively defined undirected simple graph without loops (defined as a pair (V,E) of a set V (vertex set) and a set of unordered pairs of elements of V (edge set). (Contributed by Alexander van der Vekens, 28-Aug-2017.)
 |-  G  =  { <. v ,  e >.  |  e 
 C_  { x  e.  ~P v  |  ( # `  x )  =  2 } }   =>    |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V G E 
 <->  E  C_  { x  e.  ~P V  |  ( # `  x )  =  2 } ) )
 
Theoremausisusgra 21363* The equivalence of the definitions of an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 28-Aug-2017.)
 |-  G  =  { <. v ,  e >.  |  e 
 C_  { x  e.  ~P v  |  ( # `  x )  =  2 } }   =>    |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V G E 
 <->  V USGrph  (  _I  |`  E ) ) )
 
Theoremusgraedgop 21364 An edge of an undirected simple graph as second component of an ordered pair. (Contributed by Alexander van der Vekens, 17-Aug-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
 |-  ( ( V USGrph  E  /\  X  e.  dom  E )  ->  ( ( E `
  X )  =  { M ,  N } 
 <-> 
 <. X ,  { M ,  N } >.  e.  E ) )
 
Theoremusgraf1o 21365 The edge function of an undirected simple graph without loops is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran  E )
 
Theoremusgraf1 21366 The edge function of an undirected simple graph without loops is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-> ran  E )
 
Theoremusgrass 21367 An edge is a subset of vertices, analogous to umgrass 21337. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  ( ( V USGrph  E  /\  F  e.  dom  E )  ->  ( E `  F )  C_  V )
 
Theoremusgraeq12d 21368 Equality of simple graphs without loops. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( V  =  W  /\  E  =  F )
 )  ->  ( V USGrph  E  <->  W USGrph  F ) )
 
Theoremuslisumgra 21369 An undirected simple graph with loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USLGrph  E  ->  V UMGrph  E )
 
Theoremusisuslgra 21370 An undirected simple graph without loops is an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.)
 |-  ( V USGrph  E  ->  V USLGrph  E )
 
Theoremusisumgra 21371 An undirected simple graph without loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
 |-  ( V USGrph  E  ->  V UMGrph  E )
 
Theoremusgrares 21372 A subgraph of a graph (formed by removing some edges from the original graph) is a graph, analogous to umgrares 21342. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USGrph  E  ->  V USGrph 
 ( E  |`  A ) )
 
Theoremusgra0 21373 The empty graph, with vertices but no edges, is a graph, analogous to umgra0 21343. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V  e.  W  ->  V USGrph  (/) )
 
Theoremusgra0v 21374 The empty graph with no vertices is a graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
 |-  ( (/) USGrph  E  <->  E  =  (/) )
 
Theoremuslgra1 21375 The graph with one edge, analogous to umgra1 21344. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V )
 )  ->  V USLGrph  { <. A ,  { B ,  C } >. } )
 
Theoremusgra1 21376 The graph with one edge, analogous to umgra1 21344 ( with additional assumption that  B  =/=  C since otherwise the edge is a loop!). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
 |-  ( ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V )
 )  ->  ( B  =/=  C  ->  V USGrph  { <. A ,  { B ,  C } >. } ) )
 
Theoremuslgraun 21377 If  <. V ,  E >. and  <. V ,  F >. are (simple) graphs (with loops), then  <. V ,  E  u.  F >. is a multigraph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices), analogous to umgraun 21346. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V USLGrph  E )   &    |-  ( ph  ->  V USLGrph  F )   =>    |-  ( ph  ->  V UMGrph 
 ( E  u.  F ) )
 
Theoremusgraedg2 21378 The value of the "edge function" of a graph is a set containing two elements (the vertices the corresponding edge is connecting), analogous to umgrale 21339. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
 |-  ( ( V USGrph  E  /\  X  e.  dom  E )  ->  ( # `  ( E `  X ) )  =  2 )
 
Theoremusgraedgprv 21379 In an undirected graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  ( ( V USGrph  E  /\  X  e.  dom  E )  ->  ( ( E `
  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V ) ) )
 
Theoremusgraedgrnv 21380 An edge of an undirected simple graph always connects two vertices. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
 |-  ( ( V USGrph  E  /\  { M ,  N }  e.  ran  E ) 
 ->  ( M  e.  V  /\  N  e.  V ) )
 
Theoremusgranloopv 21381 In an undirected simple graph without loops, there is no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  ( ( V USGrph  E  /\  M  e.  W ) 
 ->  ( ( E `  X )  =  { M ,  N }  ->  M  =/=  N ) )
 
Theoremusgranloop 21382* In an undirected simple graph without loops, there is no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.)
 |-  ( V USGrph  E  ->  ( E. x  e.  dom  E ( E `  x )  =  { M ,  N }  ->  M  =/=  N ) )
 
Theoremusgranloop0 21383* A simple undirected graph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
 |-  ( ( V USGrph  E  /\  U  e.  V ) 
 ->  { x  e.  dom  E  |  ( E `  x )  =  { U } }  =  (/) )
 
Theoremusgraedgrn 21384 An edge of an undirected simple graph without loops always connects two different vertices. (Contributed by Alexander van der Vekens, 2-Sep-2017.)
 |-  ( ( V USGrph  E  /\  { M ,  N }  e.  ran  E ) 
 ->  M  =/=  N )
 
Theoremusgra2edg 21385* If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
 |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c
 )  /\  ( { N ,  b }  e.  ran  E  /\  {
 c ,  N }  e.  ran  E ) ) 
 ->  E. x  e.  dom  E E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `
  x )  /\  N  e.  ( E `  y ) ) )
 
Theoremusgra2edg1 21386* If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
 |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c
 )  /\  ( { N ,  b }  e.  ran  E  /\  {
 c ,  N }  e.  ran  E ) ) 
 ->  -.  E! x  e. 
 dom  E  N  e.  ( E `  x ) )
 
Theoremusgrarnedg 21387* For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.)
 |-  ( ( V USGrph  E  /\  Y  e.  ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) )
 
Theoremusgraedg3 21388* The value of the "edge function" of a graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.)
 |-  ( ( V USGrph  E  /\  X  e.  dom  E )  ->  E. x  e.  V  E. y  e.  V  ( x  =/=  y  /\  ( E `  X )  =  { x ,  y } ) )
 
Theoremusgraedg4 21389* The value of the "edge function" of a graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.)
 |-  ( ( V USGrph  E  /\  X  e.  dom  E  /\  Y  e.  ( E `
  X ) ) 
 ->  E. y  e.  V  ( E `  X )  =  { Y ,  y } )
 
Theoremusgraedgreu 21390* The value of the "edge function" of a graph is a uniquely determined set containing two elements (the endvertices of the corresponding edge). Concretising usgraedg4 21389. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  ( ( V USGrph  E  /\  X  e.  dom  E  /\  Y  e.  ( E `
  X ) ) 
 ->  E! y  e.  V  ( E `  X )  =  { Y ,  y } )
 
Theoremusgrarnedg1 21391* For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.)
 |-  ( ( V USGrph  E  /\  E. y  e.  ran  E  y  =  ( E `
  I ) ) 
 ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  ( E `  I
 )  =  { a ,  b } ) )
 
Theoremusgra1v 21392 A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  ( { A } USGrph  E  <->  E  =  (/) )
 
Theoremusgraidx2vlem1 21393* Lemma 1 for usgraidx2v 21395. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  A  =  { x  e.  dom  E  |  N  e.  ( E `  x ) }   =>    |-  ( ( V USGrph  E  /\  Y  e.  A ) 
 ->  ( iota_ z  e.  V ( E `  Y )  =  { z ,  N } )  e.  V )
 
Theoremusgraidx2vlem2 21394* Lemma 2 for usgraidx2v 21395. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  A  =  { x  e.  dom  E  |  N  e.  ( E `  x ) }   =>    |-  ( ( V USGrph  E  /\  Y  e.  A ) 
 ->  ( I  =  (
 iota_ z  e.  V ( E `  Y )  =  { z ,  N } )  ->  ( E `  Y )  =  { I ,  N } ) )
 
Theoremusgraidx2v 21395* The mapping of indices of edges containing a given vertex into the set of vertices is 1-1. The index is mapped to the other vertex of the edge containing the vertex N. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  A  =  { x  e.  dom  E  |  N  e.  ( E `  x ) }   &    |-  F  =  ( y  e.  A  |->  (
 iota_ z  e.  V ( E `  y )  =  { z ,  N } ) )   =>    |-  ( ( V USGrph  E  /\  N  e.  V ) 
 ->  F : A -1-1-> V )
 
Theoremusgraedgleord 21396* In a graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
 |-  ( ( V USGrph  E  /\  N  e.  V ) 
 ->  ( # `  { x  e.  dom  E  |  N  e.  ( E `  x ) } )  <_  ( # `
  V ) )
 
14.1.3.2  Undirected simple graphs - examples
 
Theoremusgraexvlem 21397 Lemma for usgraexmpl 21403. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  V  =  ( { 0 ,  1 ,  2 }  u.  { 3 ,  4 } )
 
Theoremusgraex0elv 21398 Lemma 0 for usgraexmpl 21403. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  0  e.  V
 
Theoremusgraex1elv 21399 Lemma 1 for usgraexmpl 21403. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  1  e.  V
 
Theoremusgraex2elv 21400 Lemma 2 for usgraexmpl 21403. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0
 ... 4 )   =>    |-  2  e.  V
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