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Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcvmlift 23001* One of the important properties of covering maps is that any path in the base space "lifts" to a path in the covering space such that , and given a starting point in the covering space this lift is unique. The proof is contained in cvmliftlem1 22987 thru cvmliftlem15 23000. (Contributed by Mario Carneiro, 16-Feb-2015.)
CovMap

Theoremcvmfo 23002 A covering map is an onto function. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmliftiota 23003* Write out a function that is the unique lift of . (Contributed by Mario Carneiro, 16-Feb-2015.)
CovMap

Theoremcvmlift2lem1 23004* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 1-Jun-2015.)

Theoremcvmlift2lem9a 23005* Lemma for cvmlift2 23018 and cvmlift3 23030. (Contributed by Mario Carneiro, 9-Jul-2015.)
t t        CovMap                                                                t

Theoremcvmlift2lem2 23006* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2lem3 23007* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2lem4 23008* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap

Theoremcvmlift2lem5 23009* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2lem6 23010* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap                                           t

Theoremcvmlift2lem7 23011* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2lem8 23012* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 9-Mar-2015.)
CovMap

Theoremcvmlift2lem9 23013* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap                                           t t                                    t        t                                    t               t

Theoremcvmlift2lem10 23014* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap                                           t t                      t t

Theoremcvmlift2lem11 23015* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap                                                                              t t

Theoremcvmlift2lem12 23016* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap

Theoremcvmlift2lem13 23017* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2 23018* A two-dimensional version of cvmlift 23001. There is a unique lift of functions on the unit square which commutes with the covering map. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap

Theoremcvmliftphtlem 23019* Lemma for cvmliftpht 23020. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap

Theoremcvmliftpht 23020* If and are path-homotopic, then their lifts and are also path-homotopic. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap

Theoremcvmlift3lem1 23021* Lemma for cvmlift3 23030. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap        SCon       𝑛Locally PCon

Theoremcvmlift3lem2 23022* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap        SCon       𝑛Locally PCon

Theoremcvmlift3lem3 23023* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap        SCon       𝑛Locally PCon

Theoremcvmlift3lem4 23024* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap        SCon       𝑛Locally PCon

Theoremcvmlift3lem5 23025* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap        SCon       𝑛Locally PCon

Theoremcvmlift3lem6 23026* Lemma for cvmlift3 23030. (Contributed by Mario Carneiro, 9-Jul-2015.)
CovMap        SCon       𝑛Locally PCon                                          t t                                                                       t

Theoremcvmlift3lem7 23027* Lemma for cvmlift3 23030. (Contributed by Mario Carneiro, 9-Jul-2015.)
CovMap        SCon       𝑛Locally PCon                                          t t                                    t PCon

Theoremcvmlift3lem8 23028* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap        SCon       𝑛Locally PCon                                          t t

Theoremcvmlift3lem9 23029* Lemma for cvmlift2 23018. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap        SCon       𝑛Locally PCon                                          t t

Theoremcvmlift3 23030* A general version of cvmlift 23001. If is simply connected and weakly locally path-connected, then there is a unique lift of functions on which commutes with the covering map. (Contributed by Mario Carneiro, 9-Jul-2015.)
CovMap        SCon       𝑛Locally PCon

16.3.10  Undirected multigraphs

Syntaxcumg 23031 Extend class notation with undirected multigraphs.
UMGrph

Syntaxceup 23032 Extend class notation with Eulerian paths.
EulPaths

Syntaxcvdg 23033 Extend class notation with the vertex degree function.
VDeg

Definitiondf-umgra 23034* 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

Definitiondf-eupa 23035* Define the set of all Eulerian paths on an undirected multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths UMGrph

Definitiondf-vdgr 23036* Define the vertex degree function for an undirected multigraph. We have to double-count those edges that contain "twice" (i.e. self-loops), this being represented as a singleton as the edge's value. (Contributed by Mario Carneiro, 12-Mar-2015.)
VDeg

Theoremrelumgra 23037 The class of all undirected multigraphs is a relation. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremisumgra 23038* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremwrdumgra 23039* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
Word UMGrph Word

Theoremumgraf2 23040* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
UMGrph

Theoremumgraf 23041* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremumgrass 23042 An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremumgran0 23043 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremumgrale 23044 An edge has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremumgrafi 23045 An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremumgraex 23046* An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph

Theoremumgrares 23047 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 23048 The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
UMGrph

Theoremumgra1 23049 The graph with one edge. (Contributed by Mario Carneiro, 12-Mar-2015.)
UMGrph

Theoremumgraun 23050 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

Theoremreleupa 23051 The set EulPaths of all Eulerian paths on is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremiseupa 23052* The property " is an Eulerian path on the graph ". An Eulerian path is defined as bijection from the edges to a set a function into the vertices such that for each , is an edge from to . (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
EulPaths UMGrph

Theoremeupagra 23053 If an eulerian path exists, then is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths UMGrph

Theoremeupai 23054* Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupacl 23055 An Eulerian path has length , which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupaf1o 23056 The function in an Eulerian path is a bijection from a one-based sequence to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupafi 23057 Any graph with an Eulerian path is finite. (Contributed by Mario Carneiro, 7-Apr-2015.)
EulPaths

Theoremeupapf 23058 The function in an Eulerian path is a function from a zero-based finite sequence to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupaseg 23059 The -th edge in an eulerian path is the edge from to . (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremvdgrfval 23060* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.)
VDeg

Theoremvdgrval 23061* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.)
VDeg

Theoremvdgrf 23062 The vertex degree function on finite graphs is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.)
VDeg

Theoremvdgr0 23063 The degree of a vertex in an empty graph is zero, because there are no edges. (Contributed by Mario Carneiro, 12-Mar-2015.)
VDeg

Theoremvdgrun 23064 The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
UMGrph        UMGrph               VDeg VDeg VDeg

Theoremvdgr1d 23065 The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.)
VDeg

Theoremvdgr1b 23066 The vertex degree of a one-edge graph, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.)
VDeg

Theoremvdgr1c 23067 The vertex degree of a one-edge graph, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.)
VDeg

Theoremvdgr1a 23068 The vertex degree of a one-edge graph, case 1: an edge between two vertices other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.)
VDeg

Theoremeupa0 23069 There is an Eulerian path on the empty graph. (Contributed by Mario Carneiro, 7-Apr-2015.)
EulPaths

Theoremeupares 23070 The restriction of an Eulerian path to an initial segment of the path forms an Eulerian path on the subgraph consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
EulPaths                                    EulPaths

Theoremeupap1 23071 Append one path segment to an Eulerian path (enlarging the graph to add the new edge). (Contributed by Mario Carneiro, 7-Apr-2015.)
EulPaths                                    EulPaths

Theoremeupath2lem1 23072 Lemma for eupath2 23075. (Contributed by Mario Carneiro, 8-Apr-2015.)

Theoremeupath2lem2 23073 Lemma for eupath2 23075. (Contributed by Mario Carneiro, 8-Apr-2015.)

Theoremeupath2lem3 23074* Lemma for eupath2 23075. (Contributed by Mario Carneiro, 8-Apr-2015.)
EulPaths                             VDeg        VDeg

Theoremeupath2 23075* The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.)
EulPaths        VDeg

Theoremeupath 23076* A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.)
EulPaths VDeg

Theoremvdeg0i 23077 The base case for the induction for calculating the degree of a vertex. The degree of in the empty graph is . (Contributed by Mario Carneiro, 12-Mar-2015.)
VDeg

Theoremumgrabi 23078* Show that an unordered pair is a valid edge in a graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

Theoremvdegp1ai 23079* The induction step for a vertex degree calculation. If the degree of in the edge set is , then adding to the edge set, where , yields degree as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Word               VDeg                                    concat        VDeg

Theoremvdegp1bi 23080* The induction step for a vertex degree calculation. If the degree of in the edge set is , then adding to the edge set, where , yields degree . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Word               VDeg                             concat        VDeg

Theoremvdegp1ci 23081* The induction step for a vertex degree calculation. If the degree of in the edge set is , then adding to the edge set, where , yields degree . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Word               VDeg                             concat        VDeg

Theoremkonigsberg 23082 The Konigsberg Bridge problem. If is the graph on four vertices , with edges , then vertices each have degree three, and has degree five, so there are four vertices of odd degree and thus by eupath 23076 the graph cannot have an Eulerian path. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
EulPaths

16.3.11  Normal numbers

Theoremsnmlff 23083* The function from snmlval 23085 is a mapping from positive integers to real numbers in the range . (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremsnmlfval 23084* The function from snmlval 23085 maps to the relative density of in the first digits of the digit string of in base . (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremsnmlval 23085* The property " is simply normal in base ". A number is simply normal if each digit occurs in the base- digit string of with frequency (which is consistent with the expectation in an infinite random string of numbers selected from ). (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremsnmlflim 23086* If is simply normal, then the function of relative density of in the digit string converges to , i.e. the set of occurences of in the digit string has natural density . (Contributed by Mario Carneiro, 6-Apr-2015.)

16.3.12  Godel-sets of formulas

Syntaxcgoe 23087 The Godel-set of membership.

Syntaxcgna 23088 The Godel-set for the Sheffer stroke.

Syntaxcgol 23089 The Godel-set of universal quantification. (Note that this is not a wff.)

Syntaxcsat 23090 The satisfaction function.

Syntaxcfmla 23091 The formula set predicate.

Syntaxcsate 23092 The -satisfaction function.

Syntaxcprv 23093 The "proves" relation.

Definitiondf-goel 23094 Define the Godel-set of membership. Here the arguments correspond to vN and vP , so actually means v0 v1 , not . (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gona 23095 Define the Godel-set for the Sheffer stroke NAND. Here the arguments are also Godel-sets corresponding to smaller formulae. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-goal 23096 Define the Godel-set of universal quantification. Here corresponds to vN , and represents another formula, and this expression is where is the -th variable, is the code for . Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-sat 23097* Define the satisfaction predicate. This recursive construction builds up a function over wff codes and simultaneously defines the set of assignments to all variables from that makes the coded wff true in the model , where is interpreted as the binary relation on . The interpretation of the statement is that for the model , is an valuation of the variables (v0 , v1 , etc.) and is a code for a wff using that is true under the assignment . The function is defined by finite recursion; only operates on wffs of depth at most , and operates on all wffs. The coding scheme for the wffs is defined so that
• vi vj is coded as ,
• is coded as , and
• vi is coded as .

(Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-sate 23098* A simplified version of the satisfaction predicate, using the standard membership relation and eliminating the extra variable . (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-fmla 23099 Define the predicate which defines the set of valid Godel formulas. The parameter defines the maximum height of the formulas: the set is all formulas of the form or (which in our coding scheme is the set ; see df-sat 23097 for the full coding scheme), and each extra level adds to the complexity of the formulas in . is the set of all valid formulas. (Contributed by Mario Carneiro, 14-Jul-2013.)

Syntaxcgon 23100 The Godel-set of negation. (Note that this is not a wff.)

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