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Theorem List for Metamath Proof Explorer - 21301-21400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempntlemr 21301* Lemma for pntlemj 21302. (Contributed by Mario Carneiro, 7-Jun-2016.)
ψ                                    ;                                                         ;                                                                ..^

Theorempntlemj 21302* Lemma for pnt 21313. The induction step. Using pntibnd 21292, we find an interval in which is sufficiently large and has a much smaller value, (instead of our original bound ). (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;                                                                ..^

Theorempntlemi 21303* Lemma for pnt 21313. Eliminate some assumptions from pntlemj 21302. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;                                                  ..^

Theorempntlemf 21304* Lemma for pnt 21313. Add up the pieces in pntlemi 21303 to get an estimate slightly better than the naive lower bound . (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;                                           ;

Theorempntlemk 21305* Lemma for pnt 21313. Evaluate the naive part of the estimate. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlemo 21306* Lemma for pnt 21313. Combine all the estimates to establish a smaller eventual bound on . (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntleme 21307* Lemma for pnt 21313. Package up pntlemo 21306 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlem3 21308* Lemma for pnt 21313. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ                                           ψ

Theorempntlemp 21309* Lemma for pnt 21313. Wrapping up more quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                           ;

Theorempntleml 21310* Lemma for pnt 21313. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                           ;               ψ

Theorempnt3 21311 The Prime Number Theorem, version 3: the second Chebyshev function tends asymptotically to . (Contributed by Mario Carneiro, 1-Jun-2016.)
ψ

Theorempnt2 21312 The Prime Number Theorem, version 2: the first Chebyshev function tends asymptotically to . (Contributed by Mario Carneiro, 1-Jun-2016.)

Theorempnt 21313 The Prime Number Theorem: the number of prime numbers less than tends asymptotically to as goes to infinity. (Contributed by Mario Carneiro, 1-Jun-2016.)
π

13.4.13  Ostrowski's theorem

Theoremabvcxp 21314* Raising an absolute value to a power less than one yields another absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theorempadicfval 21315* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theorempadicval 21316* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theoremostth2lem1 21317* Lemma for ostth2 21336, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 21336. If a power is upper bounded by a linear term, the exponent must be less than one. Or in big-O notation, for any . (Contributed by Mario Carneiro, 10-Sep-2014.)

Theoremqrngbas 21318 The base set of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqdrng 21319 The rationals form a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrng0 21320 The zero element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrng1 21321 The unit element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrngneg 21322 The additive inverse in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrngdiv 21323 The division operation in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds        /r

Theoremqabvle 21324 By using induction on , we show a long-range inequality coming from the triangle inequality. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremqabvexp 21325 Induct the product rule abvmul 15922 to find the absolute value of a power. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostthlem1 21326* Lemma for ostth 21338. 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.)
flds        AbsVal

Theoremostthlem2 21327* Lemma for ostth 21338. Refine ostthlem1 21326 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.)
flds        AbsVal

Theoremqabsabv 21328 The regular absolute value function on the rationals is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theorempadicabv 21329* The p-adic absolute value (with arbitrary base) is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theorempadicabvf 21330* The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theorempadicabvcxp 21331* All positive powers of the p-adic absolute value are absolute values. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theoremostth1 21332* - Lemma for ostth 21338: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If is equal to on the primes, then by complete induction and the multiplicative property abvmul 15922 of the absolute value, is equal to on all the integers, and ostthlem1 21326 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2lem2 21333* Lemma for ostth2 21336. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2lem3 21334* Lemma for ostth2 21336. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2lem4 21335* Lemma for ostth2 21336. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2 21336* - Lemma for ostth 21338: regular case. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth3 21337* - Lemma for ostth 21338: p-adic case. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth 21338* Ostrowski's theorem, which classifies all absolute values on . Any such absolute value must either be the trivial absolute value , a constant exponent times the regular absolute value, or a positive exponent times the p-adic absolute value. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

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 21340 an ordered pair of a set and a function into the powerset of ( ).
An element of is called "vertex", an element of is called "edge", the function 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 21353 a graph such that is a function into the set of (proper or not proper) unordered pairs of .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 3982).
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 21371 a graph such that is a one-to-one function into the set of (proper or not proper) unordered pairs of .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 21372 a graph such that is a one-to-one function into the set of (proper) unordered pairs of .An ordered pair of two distinct sets and (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 21385.
Finite graph---a graph with finite sets and .In simple graphs, is finite if is finite, see usgrafis 21434. The number of edges is limited by (or " choose 2") with , see usgramaxsize 21501. Analogously, the number of edges of an undirected simple graph with loops is limited by . In multigraphs, however, can be infinite although is finite.
Graph of finite size---a graph with finite set , i.e. with a finite number of edges.A graph can be of finite size although is infinite.

Terms and properties of graphs:
TermReferenceDefinitionRemarks
Edge joining (two) vertices --- An edge "joins" the vertices v1, v2, ... vn ( ) if = { v1, v2, ... vn }. If , = { v1 } is a "loop", if , = { 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 joins the vertices v1, v2, ... vn ( ), then the vertices v1, v2, ... vn are called the "endvertices" of the edge .
(Two) Adjacent vertices see definition in Section I.1 in [Bollobas] p. 1/2. The vertices v1, v2, ... vn ( ) 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 ( ) 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 is the number of vertices in the graph ().
Size of a graph see definition in Section I.1 in [Bollobas] p. 3 the "size" of a graph is the number of edges in the graph ().
Neighborhood of a vertex df-nbgra 21438 resp. definition in Section I.1 in [Bollobas] p. 3 A vertex connected with a vertex by an edge is called a "neighbor" of the vertex . The set of neighbors of a vertex is called the "neighborhood" (or "open neighborhood") of the vertex . The "closed neighborhood" is the union of the (open) neighborhood of the vertex with .
Degree of a vertex df-vdgr 21670 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 21688 A vertex is called "isolated" if it is not an endvertex of any edge, thus having degree 0.
Universal vertex df-uvtx 21440 A vertex is called "universal" if it is connected with every other vertex of the graph by an edge, thus having degree .

Special kinds of graphs:
TermReferenceDefinitionRemarks
Complete graph df-cusgra 21439 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 is (or " choose 2"), see cusgrasize 21492.
Empty graph umgra0 21365 and usgra0 21395 A graph is called "empty" if it has no edges.
Null graph usgra0v 21396 A graph is called the "null graph" if it has no vertices (and therefore also no edges).
Trivial graph usgra1v 21414 A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph df-conngra 21662 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 21339 Extend class notation with undirected hypergraphs.
UHGrph

Definitiondf-uhgra 21340* 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

Theoremreluhgra 21341 The class of all undirected hypergraphs is a relation. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph

Theoremuhgrav 21342 The classes of vertices and edges of an undirected hypergraph are sets. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph

Theoremisuhgra 21343 The property of being an undirected hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph

Theoremuhgraf 21344 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.)
UHGrph

Theoremuhgrafun 21345 The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph

Theoremuhgrass 21346 An edge is a subset of vertices, analogous to umgrass 21359. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph

Theoremuhgraeq12d 21347 Equality of hypergraphs. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph UHGrph

Theoremuhgrares 21348 A subgraph of a hypergraph (formed by removing some edges from the original graph) is a hypergraph, analogous to umgrares 21364. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
UHGrph UHGrph

Theoremuhgra0 21349 The empty graph, with vertices but no edges, is a hypergraph, analogous to umgra0 21365. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
UHGrph

Theoremuhgra0v 21350 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

Theoremuhgraun 21351 If and are hypergraphs, then 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 21368. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
UHGrph        UHGrph        UHGrph

14.1.2  Undirected multigraphs

Syntaxcumg 21352 Extend class notation with undirected multigraphs.
UMGrph

Definitiondf-umgra 21353* Define the class of all undirected multigraphs. A multigraph is a pair where is a function into subsets of 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

Theoremrelumgra 21354 The class of all undirected multigraphs is a relation. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremisumgra 21355* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremwrdumgra 21356* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
Word UMGrph Word

Theoremumgraf2 21357* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
UMGrph

Theoremumgraf 21358* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremumgrass 21359 An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremumgran0 21360 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremumgrale 21361 An edge has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremumgrafi 21362 An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremumgraex 21363* An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremumgrares 21364 A subgraph of a graph (formed by removing some edges from the original graph) is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
UMGrph UMGrph

Theoremumgra0 21365 The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
UMGrph

Theoremumgra1 21366 The graph with one edge. (Contributed by Mario Carneiro, 12-Mar-2015.)
UMGrph

Theoremumisuhgra 21367 An undirected multigraph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
UMGrph UHGrph

Theoremumgraun 21368 If and are graphs, then is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.)
UMGrph        UMGrph        UMGrph

14.1.3  Undirected simple graphs

14.1.3.1  Undirected simple graphs - basics

Syntaxcuslg 21369 Extend class notation with undirected (simple) graphs with loops.
USLGrph

Syntaxcusg 21370 Extend class notation with undirected (simple) graphs (without loops).
USGrph

Definitiondf-uslgra 21371* Define the class of all undirected simple graphs with loops. An undirected simple graph with loops is a special undirected multigraph where is an injective (one-to-one) function into subsets of 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

Definitiondf-usgra 21372* Define the class of all undirected simple graphs without loops. An undirected simple graph without loops is a special undirected simple graph where is an injective (one-to-one) function into subsets of 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

Theoremreluslgra 21373 The class of all undirected simple graph with loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USLGrph

Theoremrelusgra 21374 The class of all undirected simple graph without loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USGrph

Theoremuslgrav 21375 The classes of vertices and edges of an undirected simple graph with loops are sets. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
USLGrph

Theoremusgrav 21376 The classes of vertices and edges of an undirected simple graph without loops are sets. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
USGrph

Theoremisuslgra 21377* The property of being an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USLGrph

Theoremisusgra 21378* The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USGrph

Theoremuslgraf 21379* 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.)
USLGrph

Theoremusgraf 21380* 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.)
USGrph

Theoremisusgra0 21381* The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
USGrph

Theoremusgraf0 21382* 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.)
USGrph

Theoremusgrafun 21383 The edge function of an undirected simple graph without loops is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017.)
USGrph

Theoremisausgra 21384* 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.)

Theoremausisusgra 21385* The equivalence of the definitions of an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 28-Aug-2017.)
USGrph

Theoremusgraedgop 21386 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.)
USGrph

Theoremusgraf1o 21387 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.)
USGrph

Theoremusgraf1 21388 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.)
USGrph

Theoremusgrass 21389 An edge is a subset of vertices, analogous to umgrass 21359. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
USGrph

Theoremusgraeq12d 21390 Equality of simple graphs without loops. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
USGrph USGrph

Theoremuslisumgra 21391 An undirected simple graph with loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USLGrph UMGrph

Theoremusisuslgra 21392 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.)
USGrph USLGrph

Theoremusisumgra 21393 An undirected simple graph without loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
USGrph UMGrph

Theoremusgrares 21394 A subgraph of a graph (formed by removing some edges from the original graph) is a graph, analogous to umgrares 21364. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USGrph USGrph

Theoremusgra0 21395 The empty graph, with vertices but no edges, is a graph, analogous to umgra0 21365. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USGrph

Theoremusgra0v 21396 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

Theoremuslgra1 21397 The graph with one edge, analogous to umgra1 21366. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USLGrph

Theoremusgra1 21398 The graph with one edge, analogous to umgra1 21366 ( with additional assumption that 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.)
USGrph

Theoremuslgraun 21399 If and are (simple) graphs (with loops), then 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 21368. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USLGrph        USLGrph        UMGrph

Theoremusgraedg2 21400 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 21361. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
USGrph

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