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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeupath2lem1 23901 Lemma for eupath2 23904. (Contributed by Mario Carneiro, 8-Apr-2015.)

Theoremeupath2lem2 23902 Lemma for eupath2 23904. (Contributed by Mario Carneiro, 8-Apr-2015.)

Theoremeupath2lem3 23903* Lemma for eupath2 23904. (Contributed by Mario Carneiro, 8-Apr-2015.)
EulPaths                             VDeg        VDeg

Theoremeupath2 23904* 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 23905* 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 23906 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 23907* 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 23908* 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 23909* 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 23910* 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 23911 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 23905 the graph cannot have an Eulerian path. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
EulPaths

18.4.11  Normal numbers

Theoremsnmlff 23912* The function from snmlval 23914 is a mapping from positive integers to real numbers in the range . (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremsnmlfval 23913* The function from snmlval 23914 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 23914* 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 23915* 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.)

18.4.12  Godel-sets of formulas

Syntaxcgoe 23916 The Godel-set of membership.

Syntaxcgna 23917 The Godel-set for the Sheffer stroke.

Syntaxcgol 23918 The Godel-set of universal quantification. (Note that this is not a wff.)

Syntaxcsat 23919 The satisfaction function.

Syntaxcfmla 23920 The formula set predicate.

Syntaxcsate 23921 The -satisfaction function.

Syntaxcprv 23922 The "proves" relation.

Definitiondf-goel 23923 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 23924 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 23925 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 23926* 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 23927* 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 23928 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 23926 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 23929 The Godel-set of negation. (Note that this is not a wff.)

Syntaxcgoa 23930 The Godel-set of conjunction.

Syntaxcgoi 23931 The Godel-set of implication.

Syntaxcgoo 23932 The Godel-set of disjunction.

Syntaxcgob 23933 The Godel-set of equivalence.

Syntaxcgoq 23934 The Godel-set of equality.

Syntaxcgox 23935 The Godel-set of existential quantification. (Note that this is not a wff.)

Definitiondf-gonot 23936 Define the Godel-set of negation. Here the argument is also a Godel-set corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-goan 23937* Define the Godel-set of conjunction. Here the arguments and are also Godel-sets corresponding to smaller formulae. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-goim 23938* Define the Godel-set of implication. Here the arguments and are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-goor 23939* Define the Godel-set of disjunction. Here the arguments and are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gobi 23940* Define the Godel-set of equivalence. Here the arguments and are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-goeq 23941* Define the Godel-set of equality. Here the arguments correspond to vN and vP , so actually means v0 v1 , not . Here we use the trick mentioned in ax-ext 2264 to introduce equality as a defined notion in terms of . The expression max here is a convenient way of getting a dummy variable distinct from and . (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-goex 23942 Define the Godel-set of existential 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-prv 23943* Define the "proves" relation on a set. A wff is true in a model if for every valuation , the interpretation of the wff using the membership relation on is true. (Contributed by Mario Carneiro, 14-Jul-2013.)

18.4.13  Models of ZF

Syntaxcgze 23944 The Axiom of Extensionality.

Syntaxcgzr 23945 The Axiom Scheme of Replacement.

Syntaxcgzp 23946 The Axiom of Power Sets.

Syntaxcgzu 23947 The Axiom of Unions.

Syntaxcgzg 23948 The Axiom of Regularity.

Syntaxcgzi 23949 The Axiom of Infinity.

Syntaxcgzf 23950 The set of models of ZF.

Definitiondf-gzext 23951 The Godel-set version of the Axiom of Extensionality. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gzrep 23952 The Godel-set version of the Axiom Scheme of Replacement. Since this is a scheme and not a single axiom, it manifests as a function on wffs, each giving rise to a different axiom. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gzpow 23953 The Godel-set version of the Axiom of Power Sets. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gzun 23954 The Godel-set version of the Axiom of Unions. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gzreg 23955 The Godel-set version of the Axiom of Regularity. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gzinf 23956 The Godel-set version of the Axiom of Infinity. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gzf 23957* Define the class of all (transitive) models of ZF. (Contributed by Mario Carneiro, 14-Jul-2013.)

18.4.14  Splitting fields

Syntaxcitr 23958 Integral subring of a ring.
IntgRing

Syntaxccpms 23959 Completion of a metric space.
cplMetSp

Syntaxchlb 23960 Embeddings for a direct limit.
HomLimB

Syntaxchlim 23961 Direct limit structure.
HomLim

Syntaxcpfl 23962 Polynomial extension field.
polyFld

Syntaxcsf1 23963 Splitting field for a single polynomial (auxiliary).
splitFld1

Syntaxcsf 23964 Splitting field for a finite set of polynomials.
splitFld

Syntaxcpsl 23965 Splitting field for a sequence of polynomials.
polySplitLim

Definitiondf-irng 23966* Define the subring of elements of integral over in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
IntgRing Monic1ps

Definitiondf-cplmet 23967* A function which completes the given metric space. (Contributed by Mario Carneiro, 2-Dec-2014.)
cplMetSp s s s sSet

Definitiondf-homlimb 23968* The input to this function is a sequence (on ) of homomorphisms . The resulting structure is the direct limit of the direct system so defined. This function returns the pair where is the terminal object and is a sequence of functions such that and . (Contributed by Mario Carneiro, 2-Dec-2014.)
HomLimB

Definitiondf-homlim 23969* The input to this function is a sequence (on ) of structures and homomorphisms . The resulting structure is the direct limit of the direct system so defined, and maintains any structures that were present in the original objects. TODO: generalize to directed sets? (Contributed by Mario Carneiro, 2-Dec-2014.)
HomLim HomLimB

Definitiondf-plfl 23970* Define the field extension that augments a field with the root of the given irreducible polynomial, and extends the norm if one exists and the extension is unique. (Contributed by Mario Carneiro, 2-Dec-2014.)
polyFld Poly1 RSpan ~QG s ~QG toNrmGrp AbsVal sSet deg1 deg1

Definitiondf-sfl1 23971* Temporary construction for the splitting field of a polynomial. The inputs are a field and a polynomial that we want to split, along with a tuple in the same format as the output. The output is a tuple where is the splitting field and is an injective homomorphism from the original field .

The function works by repeatedly finding the smallest monic irreducible factor, and extending the field by that factor using the polyFld construction. We keep track of a total order in each of the splitting fields so that we can pick an element definably without needing global choice. (Contributed by Mario Carneiro, 2-Dec-2014.)

splitFld1 Poly1 mPoly Monic1p Irred r deg1 polyFld deg1

Definitiondf-sfl 23972* Define the splitting field of a finite collection of polynomials, given a total ordered base field. The output is a tuple where is the totally ordered splitting field and is an injective homomorphism from the original field . (Contributed by Mario Carneiro, 2-Dec-2014.)
splitFld splitFld1

Definitiondf-psl 23973* Define the direct limit of an increasing sequence of fields produced by pasting together the splitting fields for each sequence of polynomials. That is, given a ring , a strict order on , and a sequence of finite sets of polynomials to split, we construct the direct limit system of field extensions by splitting one set at a time and passing the resulting construction to HomLim. (Contributed by Mario Carneiro, 2-Dec-2014.)
polySplitLim splitFld HomLim

18.4.15  p-adic number fields

Syntaxczr 23974 Integral elements of a ring.
ZRing

Syntaxcgf 23975 Galois finite field.
GF

Syntaxcgfo 23976 Galois limit field.
GF

Syntaxceqp 23977 Equivalence relation for df-qp 23988.
~Qp

Syntaxcrqp 23978 Equivalence relation representatives for df-qp 23988.
/Qp

Syntaxcqp 23979 The set of -adic rational numbers.
Qp

Syntaxczp 23980 The set of -adic integers. (Not to be confused with czn 16454.)
Zp

Syntaxcqpa 23981 Algebraic completion of the -adic rational numbers.
_Qp

Syntaxccp 23982 Metric completion of _Qp.
Cp

Definitiondf-zrng 23983 Define the subring of integral elements in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
ZRing IntgRing RHom

Definitiondf-gf 23984* Define the Galois finite field of order . (Contributed by Mario Carneiro, 2-Dec-2014.)
GF ℤ/n splitFld Poly1 var1 .gmulGrp

Definitiondf-gfoo 23985* Define the Galois field of order , as a direct limit of the Galois finite fields. (Contributed by Mario Carneiro, 2-Dec-2014.)
GF ℤ/n polySplitLim Poly1 var1 .gmulGrp

Definitiondf-eqp 23986* Define an equivalence relation on -indexed sequences of integers such that two sequences are equivalent iff the difference is equivalent to zero, and a sequence is equivalent to zero iff the sum is a multiple of for every . (Contributed by Mario Carneiro, 2-Dec-2014.)
~Qp

Definitiondf-rqp 23987* There is a unique element of ~Qp -equivalent to any element of , if the sequences are zero for sufficiently large negative values; this function selects that element. (Contributed by Mario Carneiro, 2-Dec-2014.)
/Qp ~Qp

Definitiondf-qp 23988* Define the -adic completion of the rational numbers, as a normed field structure with a total order (that is not compatible with the operations). (Contributed by Mario Carneiro, 2-Dec-2014.)
Qp /Qp /Qp toNrmGrp

Definitiondf-zp 23989 Define the -adic integers, as a subset of the -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
Zp ZRing Qp

Definitiondf-qpa 23990* Define the completion of the -adic rationals. Here we simply define it as the splitting field of a dense sequence of polynomials (using as the -th set the collection of polynomials with degree less than and with coefficients ). Krasner's lemma will then show that all monic polynomials have splitting fields isomorphic to a sufficiently close Eisenstein polynomial from the list, and unramified extensions are generated by the polynomial , which is in the list. Thus every finite extension of Qp is a subfield of this field extension, so it is algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.)
_Qp Qp polySplitLim Poly1 deg1 coe1

Definitiondf-cp 23991 Define the metric completion of the algebraic completion of the -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
Cp cplMetSp _Qp

18.5  Mathbox for Paul Chapman

18.5.1  Group homomorphism and isomorphism

Theoremghomgrpilem1 23992 Lemma for ghomgrpi 23994. (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom               GId                     GId

Theoremghomgrpilem2 23993 Lemma for ghomgrpi 23994. (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom               GId                     GId

Theoremghomgrpi 23994 The image of a group homomorphism from to is a subgroup of (inference version). (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom

Theoremghomsn 23995 The endomorphism of the trivial group. (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom

Theoremghomgrplem 23996 Lemma for ghomgrp 23997. (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom

Theoremghomgrp 23997 The image of a group homomorphism from to is a subgroup of . (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom

Theoremghomfo 23998 A group homomorphism maps onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
GrpOpHom

Theoremghomcl 23999 Closure of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.)
GrpOpHom

Theoremghomgsg 24000 A group homomorphism from to is also a group homomorphism from to its image in . (Contributed by Paul Chapman, 3-Mar-2008.)
GrpOpHom GrpOpHom

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