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Theorem List for Metamath Proof Explorer - 25101-25200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremghomgrp 25101 The image of a group homomorphism from to is a subgroup of . (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom

Theoremghomfo 25102 A group homomorphism maps onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
GrpOpHom

Theoremghomcl 25103 Closure of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.)
GrpOpHom

Theoremghomgsg 25104 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 25105* Lemma for ghomf1o 25106. (Contributed by Paul Chapman, 3-Mar-2008.)
GId       GId              GrpOpHom

Theoremghomf1o 25106* Two ways of saying a group homomorphism is 1-1-onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
GId       GId       GrpOpHom

Theoremelgiso 25107 Membership in the set of group isomorphisms from to . (Contributed by Paul Chapman, 25-Feb-2008.)
GrpOpHom

19.5.2  Real and complex numbers (cont.)

Theoremclimuzcnv 25108* Utility lemma to convert between and in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.)

Theoremsinccvglem 25109* as (real) . (Contributed by Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, 21-May-2014.)

Theoremsinccvg 25110* as (real) . (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)

Theoremcircum 25111* 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.)

19.5.3  Miscellaneous theorems

Theoremelfzm12 25112 Membership in a curtailed finite sequence of integers. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremnn0seqcvg 25113* 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 25114 Identity law for modulo restricted to integers. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremmodaddabs 25115 Absorbtion law for modulo. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremelfzp1b 25116 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 25117 Division of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2sqlei 25118 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2sqlti 25119 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2sqle 25120 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2sqlt 25121 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2difi 25122 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2difabsi 25123 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

19.6  Mathbox for Drahflow

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

Theoremsbcung 25124* Distribution of class substitution over union of two classes. (Contributed by Drahflow, 23-Sep-2015.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremsbcuni 25125* Distribution of class substitution over union of two classes, inference version. (Contributed by Drahflow, 23-Sep-2015.)

Theoremsbcopg 25126* Distribution of class substitution over ordered pairs. (Contributed by Drahflow, 25-Sep-2015.) (Revised by Mario Carneiro, 29-Oct-2015.)

Syntaxcrelexp 25127 Extend class notation to include relation exponentiation.

Definitiondf-relexp 25128* Definition of repeated composition of a relation with itself, aka relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexp0 25129 A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexpsucr 25130 A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexp1 25131 A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexpsucl 25132 A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexpcnv 25133 Distributivity of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexprel 25134 The exponentiation of a relation is a relation. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexpdm 25135 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexprn 25136 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexpfld 25137 The field of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexpadd 25138 Relation composition becomes addition under exponentiation. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexpindlem 25139* Principle of transitive induction, finite and non-class version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.)

Theoremrelexpind 25140* Principle of transitive induction, finite version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.)

Syntaxcrtrcl 25141 Extend class notation with recursively defined reflexive, transitive closure.
rec

Definitiondf-rtrclrec 25142* The reflexive, transitive closure of a relation constructed as the union of all finite exponentiations. (Contributed by Drahflow, 12-Nov-2015.)
rec

Theoremdfrtrclrec2 25143* If two elements are connected by a reflexive, transitive closure, then they are connected via instances the relation, for some . (Contributed by Drahflow, 12-Nov-2015.)
rec

Theoremrtrclreclem.refl 25144 The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.)
rec

Theoremrtrclreclem.subset 25145 The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015.)
rec

Theoremrtrclreclem.trans 25146 The reflexive, transitive closure is indeed transitive. (Contributed by Drahflow, 12-Nov-2015.)
rec rec rec

Theoremrtrclreclem.min 25147* The reflexive, transitive closure of is the smallest reflexive, transitive relation which contains and the identity. (Contributed by Drahflow, 12-Nov-2015.)
rec

Theoremdfrtrcl2 25148 The two definitions and rec of the reflexive, transitive closure coincide if is indeed a relation. (Contributed by Drahflow, 12-Nov-2015.)
rec

Theoremrtrclind 25149* Principle of transitive induction. The first four hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.)

19.7  Mathbox for Scott Fenton

19.7.1  ZFC Axioms in primitive form

Theoremaxextprim 25150 ax-ext 2417 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)

Theoremaxrepprim 25151 ax-rep 4320 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)

Theoremaxunprim 25152 ax-un 4701 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)

Theoremaxpowprim 25153 ax-pow 4377 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)

Theoremaxregprim 25154 ax-reg 7560 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)

Theoremaxinfprim 25155 ax-inf 7593 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 13-Oct-2010.)

Theoremaxacprim 25156 ax-ac 8339 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 26-Oct-2010.)

19.7.2  Untangled classes

Theoremuntelirr 25157* We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 25419). Using this concept, we can avoid a lot of the uses of the Axiom of Regularity. Here, we prove a series of properties of untanged classes. First, we prove that an untangled class is not a member of itself. (Contributed by Scott Fenton, 28-Feb-2011.)

Theoremuntuni 25158* The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.)

Theoremuntsucf 25159* If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremunt0 25160 The null set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremuntint 25161* If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)

Theoremefrunt 25162* If is well-founded by , then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)

Theoremuntangtr 25163* A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.)

19.7.3  Extra propositional calculus theorems

Theorem3orel1 25164 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)

Theorem3orel2 25165 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3orel3 25166 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)

Theorem3pm3.2ni 25167 Triple negated disjuntion introduction. (Contributed by Scott Fenton, 20-Apr-2011.)

Theorem3jaodd 25168 Double deduction form of 3jaoi 1247. (Contributed by Scott Fenton, 20-Apr-2011.)

Theorem3orit 25169 Closed form of 3ori 1244, (Contributed by Scott Fenton, 20-Apr-2011.)

Theorem3mix1d 25170 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)

Theorem3mix2d 25171 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)

Theorem3mix3d 25172 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)

Theorembiimpexp 25173 A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)

Theorem3orel13 25174 Elimination of two disjuncts in a triple disjunction. (Contributed by Scott Fenton, 9-Jun-2011.)

19.7.4  Misc. Useful Theorems

Theoremnepss 25175 Two classes are inequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)

Theorem3ccased 25176 Triple disjunction form of ccased 914. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdfso3 25177* Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.)

Theorembrtpid1 25178 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)

Theorembrtpid2 25179 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)

Theorembrtpid3 25180 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)

Theoremceqsrexv2 25181* Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.)

Theoremiota5f 25182* A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)

Theoremdfid4 25183 The identity function using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.)

19.7.5  Properties of reals and complexes

Theoremsqdivzi 25184 Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.)

Theoremdivelunit 25185 A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)

Theorempm2.61iine 25186 Equality version of pm2.61ii 159. (Contributed by Scott Fenton, 13-Jun-2013.)

Theoremdedekind 25187* The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 9068 with appropriate adjustments, states that, if completely preceeds , then there is some number separating the two of them. (Contributed by Scott Fenton, 13-Jun-2013.)

Theoremdedekindle 25188* The Dedekind cut theorem, with the hypothesis weakened to only require non-strict less than. (Contributed by Scott Fenton, 2-Jul-2013.)

Theoremmulcan1g 25189 A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)

Theoremmulcan2g 25190 A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)

Theoremmulge0b 25191 A condition for multiplication to be non-negative. (Contributed by Scott Fenton, 25-Jun-2013.)

Theoremmulle0b 25192 A condition for multiplication to be non-positive. (Contributed by Scott Fenton, 25-Jun-2013.)

Theoremmulsuble0b 25193 A condition for multiplication of subtraction to be non-positive. (Contributed by Scott Fenton, 25-Jun-2013.)

Theoremrelin01 25194 An interval law for less than or equal. (Contributed by Scott Fenton, 27-Jun-2013.)

Theoremsubdivcomb1 25195 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)

Theoremsubdivcomb2 25196 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)

Theoremsubeqrev 25197 Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)

Theoremfznatpl1 25198 Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)

Theoremsupfz 25199 The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)

Theoreminffz 25200 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)

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