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Theorem List for Metamath Proof Explorer - 23101-23200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcgoa 23101 The Godel-set of conjunction.

Syntaxcgoi 23102 The Godel-set of implication.

Syntaxcgoo 23103 The Godel-set of disjunction.

Syntaxcgob 23104 The Godel-set of equivalence.

Syntaxcgoq 23105 The Godel-set of equality.

Syntaxcgox 23106 The Godel-set of existential quantification. (Note that this is not a wff.)

Definitiondf-gonot 23107 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 23108* 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 23109* 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 23110* 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 23111* 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 23112* 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 2234 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 23113 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 23114* 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.)

16.3.13  Models of ZF

Syntaxcgze 23115 The Axiom of Extensionality.

Syntaxcgzr 23116 The Axiom Scheme of Replacement.

Syntaxcgzp 23117 The Axiom of Power Sets.

Syntaxcgzu 23118 The Axiom of Unions.

Syntaxcgzg 23119 The Axiom of Regularity.

Syntaxcgzi 23120 The Axiom of Infinity.

Syntaxcgzf 23121 The set of models of ZF.

Definitiondf-gzext 23122 The Godel-set version of the Axiom of Extensionality. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gzrep 23123 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 23124 The Godel-set version of the Axiom of Power Sets. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gzun 23125 The Godel-set version of the Axiom of Unions. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gzreg 23126 The Godel-set version of the Axiom of Regularity. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gzinf 23127 The Godel-set version of the Axiom of Infinity. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gzf 23128* Define the class of all (transitive) models of ZF. (Contributed by Mario Carneiro, 14-Jul-2013.)

16.3.14  Splitting fields

Syntaxcitr 23129 Integral subring of a ring.
IntgRing

Syntaxccpms 23130 Completion of a metric space.
cplMetSp

Syntaxchlb 23131 Embeddings for a direct limit.
HomLimB

Syntaxchlim 23132 Direct limit structure.
HomLim

Syntaxcpfl 23133 Polynomial extension field.
polyFld

Syntaxcsf1 23134 Splitting field for a single polynomial (auxiliary).
splitFld1

Syntaxcsf 23135 Splitting field for a finite set of polynomials.
splitFld

Syntaxcpsl 23136 Splitting field for a sequence of polynomials.
polySplitLim

Definitiondf-irng 23137* Define the subring of elements of integral over in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
IntgRing Monic1ps

Definitiondf-cplmet 23138* A function which completes the given metric space. (Contributed by Mario Carneiro, 2-Dec-2014.)
cplMetSp s s s sSet

Definitiondf-homlimb 23139* 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 23140* 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 23141* 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 23142* 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 23143* 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 23144* 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

Syntaxczr 23145 Integral elements of a ring.
ZRing

Syntaxcgf 23146 Galois finite field.
GF

Syntaxcgfo 23147 Galois limit field.
GF

Syntaxceqp 23148 Equivalence relation for df-qp 23159.
~Qp

Syntaxcrqp 23149 Equivalence relation representatives for df-qp 23159.
/Qp

Syntaxcqp 23150 The set of -adic rational numbers.
Qp

Syntaxczp 23151 The set of -adic integers. (Not to be confused with czn 16286.)
Zp

Syntaxcqpa 23152 Algebraic completion of the -adic rational numbers.
_Qp

Syntaxccp 23153 Metric completion of _Qp.
Cp

Definitiondf-zrng 23154 Define the subring of integral elements in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
ZRing IntgRing RHom

Definitiondf-gf 23155* Define the Galois finite field of order . (Contributed by Mario Carneiro, 2-Dec-2014.)
GF ℤ/n splitFld Poly1 var1 .gmulGrp

Definitiondf-gfoo 23156* 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 23157* 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 23158* 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 23159* 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 23160 Define the -adic integers, as a subset of the -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
Zp ZRing Qp

Definitiondf-qpa 23161* 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 23162 Define the metric completion of the algebraic completion of the -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
Cp cplMetSp _Qp

16.4  Mathbox for Paul Chapman

16.4.1  Group homomorphism and isomorphism

Theoremghomgrpilem1 23163 Lemma for ghomgrpi 23165. (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom               GId                     GId

Theoremghomgrpilem2 23164 Lemma for ghomgrpi 23165. (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom               GId                     GId

Theoremghomgrpi 23165 The image of a group homomorphism from to is a subgroup of (inference version). (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom

Theoremghomsn 23166 The endomorphism of the trivial group. (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom

Theoremghomgrplem 23167 Lemma for ghomgrp 23168. (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom

Theoremghomgrp 23168 The image of a group homomorphism from to is a subgroup of . (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom

Theoremghomfo 23169 A group homomorphism maps onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
GrpOpHom

Theoremghomcl 23170 Closure of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.)
GrpOpHom

Theoremghomgsg 23171 A group homomorphism from to is also a group homomorphism from to its image in . (Contributed by Paul Chapman, 3-Mar-2008.)
GrpOpHom GrpOpHom

Theoremghomf1olem 23172* Lemma for ghomf1o 23173. (Contributed by Paul Chapman, 3-Mar-2008.)
GId       GId              GrpOpHom

Theoremghomf1o 23173* Two ways of saying a group homomorphism is 1-1-onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
GId       GId       GrpOpHom

Theoremelgiso 23174 Membership in the set of group isomorphisms from to . (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom

16.4.2  Real and complex numbers (cont.)

Theoremclimuzcnv 23175* Utility lemma to convert between and in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.)

Theoremsinccvglem 23176* as (real) . (Contributed by Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, 21-May-2014.)

Theoremsinccvg 23177* as (real) . (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)

Theoremcircum 23178* The circumference of a circle of radius , defined as the limit as of the perimeter of an inscribed n-sided isogons, is . (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)

16.4.3  Miscellaneous theorems

Theoremelfzm12 23179 Membership in a curtailed finite sequence of integers. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremnn0seqcvg 23180* A strictly-decreasing nonnegative integer sequence with initial term reaches zero by the th term. Inference version. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremzmodid2 23181 Identity law for modulo restricted to integers. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremmodaddabs 23182 Absorbtion law for modulo. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremelfzp1b 23183 An integer is a member of a 0-based finite set of sequential integers iff its successor is a member of the corresponding 1-based set. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremlediv2aALT 23184 Division of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2sqlei 23185 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2sqlti 23186 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2sqle 23187 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2sqlt 23188 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2difi 23189 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2difabsi 23190 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

16.5  Mathbox for Drahflow

This is the mathbox of Jens-Wolfhard Schicke-Uffmann, reachable at drahflow@gmx.de / drahflow.name

Theoremsbcung 23191* Distribution of class substitution over union of two classes. (Contributed by Drahflow, 23-Sep-2015.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremsbcuni 23192* Distribution of class substitution over union of two classes, inference version. (Contributed by Drahflow, 23-Sep-2015.)

Theoremsbcopg 23193* Distribution of class substitution over ordered pairs. (Contributed by Drahflow, 25-Sep-2015.) (Revised by Mario Carneiro, 29-Oct-2015.)

Syntaxcrelexp 23194 Extend class notation to include relation exponentiation.

Definitiondf-relexp 23195* Definition of repeated composition of a relation with itself, aka relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexp0 23196 A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexpsucr 23197 A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexp1 23198 A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexpsucl 23199 A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexpcnv 23200 Distributivity of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.)

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