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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremrelexpexOLD 23201 Obsolete; use ovex 5735 instead - NM 5-Apr-2016. The exponentiation of a relation exists. (Contributed by Drahflow, 12-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( R ^ r N )  e.  _V
 
Theoremrelexprel 23202 The exponentiation of a relation is a relation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  Rel  ( R ^ r N ) ) )
 
Theoremrelexpdm 23203 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  dom  (  R ^ r N ) 
 C_  U. U. R ) )
 
Theoremrelexprn 23204 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  ran  (  R ^ r N ) 
 C_  U. U. R ) )
 
Theoremrelexpfld 23205 The field of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  U. U. ( R ^ r N )  C_  U. U. R ) )
 
Theoremrelexpadd 23206 Relation composition becomes addition under exponentiation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (
 ( N  e.  NN0  /\  M  e.  NN0 )  ->  ( ( R ^
 r N )  o.  ( R ^ r M ) )  =  ( R ^ r
 ( N  +  M ) ) ) )
 
Theoremrelexpindlem 23207* 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.)
 |-  ( et  ->  Rel  R )   &    |-  ( et  ->  R  e.  _V )   &    |-  ( et  ->  S  e.  _V )   &    |-  ( i  =  S  ->  ( ph  <->  ch ) )   &    |-  ( i  =  x  ->  ( ph  <->  ps ) )   &    |-  ( i  =  j  ->  ( ph  <->  th ) )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  ( j R x  ->  ( th  ->  ps )
 ) )   =>    |-  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
 r n ) x 
 ->  ps ) ) )
 
Theoremrelexpind 23208* 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.)
 |-  ( et  ->  Rel  R )   &    |-  ( et  ->  R  e.  _V )   &    |-  ( et  ->  S  e.  _V )   &    |-  ( et  ->  X  e.  _V )   &    |-  (
 i  =  S  ->  (
 ph 
 <->  ch ) )   &    |-  (
 i  =  x  ->  ( ph  <->  ps ) )   &    |-  (
 i  =  j  ->  ( ph  <->  th ) )   &    |-  ( x  =  X  ->  ( ps  <->  ta ) )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  ( j R x  ->  ( th  ->  ps )
 ) )   =>    |-  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
 r n ) X 
 ->  ta ) ) )
 
Syntaxcrtrcl 23209 Extend class notation with recursively defined reflexive, transitive closure.
 class  t *rec
 
Definitiondf-rtrclrec 23210* The reflexive, transitive closure of a relation constructed as the union of all finite exponentiations. (Contributed by Drahflow, 12-Nov-2015.)
 |-  t *rec  =  ( r  e. 
 _V  |->  U_ n  e.  NN0  ( r ^ r n ) )
 
Theoremdfrtrclrec2 23211* If two elements are connected by a reflexive, transitive closure, then they are connected via  n instances the relation, for some  n. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( A ( t *rec `  R ) B  <->  E. n  e.  NN0  A ( R ^ r n ) B ) )
 
Theoremrtrclreclem.refl 23212 The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (  _I  |`  U. U. R )  C_  ( t *rec `  R ) )
 
Theoremrtrclreclem.subset 23213 The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  R  C_  ( t *rec `  R ) )
 
Theoremrtrclreclem.trans 23214 The reflexive, transitive closure is indeed transitive. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (
 ( t *rec `  R )  o.  (
 t *rec `  R ) )  C_  ( t *rec `  R )
 )
 
Theoremrtrclreclem.min 23215* The reflexive, transitive closure of  R is the smallest reflexive, transitive relation which contains  R and the identity. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  A. s
 ( ( (  _I  |`  ( dom  R  u.  ran 
 R ) )  C_  s  /\  R  C_  s  /\  ( s  o.  s
 )  C_  s )  ->  ( t *rec `  R )  C_  s ) )
 
Theoremdfrtrcl2 23216 The two definitions  t * and  t
*rec of the reflexive, transitive closure coincide if  R is indeed a relation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (
 t * `  R )  =  ( t *rec `  R ) )
 
Theoremrtrclind 23217* 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.)
 |-  ( et  ->  Rel  R )   &    |-  ( et  ->  R  e.  _V )   &    |-  ( et  ->  S  e.  _V )   &    |-  ( et  ->  X  e.  _V )   &    |-  (
 i  =  S  ->  (
 ph 
 <->  ch ) )   &    |-  (
 i  =  x  ->  ( ph  <->  ps ) )   &    |-  (
 i  =  j  ->  ( ph  <->  th ) )   &    |-  ( x  =  X  ->  ( ps  <->  ta ) )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  ( j R x  ->  ( th  ->  ps )
 ) )   =>    |-  ( et  ->  ( S ( t * `
  R ) X 
 ->  ta ) )
 
16.6  Mathbox for Scott Fenton
 
16.6.1  ZFC Axioms in primitive form
 
Theoremaxextprim 23218 ax-ext 2234 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
 |-  -.  A. x  -.  ( ( x  e.  y  ->  x  e.  z )  ->  ( ( x  e.  z  ->  x  e.  y )  ->  y  =  z ) )
 
Theoremaxrepprim 23219 ax-rep 4028 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
 |-  -.  A. x  -.  ( -. 
 A. y  -.  A. z ( ph  ->  z  =  y )  ->  A. z  -.  (
 ( A. y  z  e.  x  ->  -.  A. x ( A. z  x  e.  y  ->  -.  A. y ph ) )  ->  -.  ( -.  A. x ( A. z  x  e.  y  ->  -.  A. y ph )  ->  A. y  z  e.  x ) ) )
 
Theoremaxunprim 23220 ax-un 4403 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
 |-  -.  A. x  -.  A. y
 ( -.  A. x ( y  e.  x  ->  -.  x  e.  z
 )  ->  y  e.  x )
 
Theoremaxpowprim 23221 ax-pow 4082 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
 |-  ( A. x  -.  A. y
 ( A. x ( -. 
 A. z  -.  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) 
 ->  x  =  y
 )
 
Theoremaxregprim 23222 ax-reg 7190 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
 |-  ( x  e.  y  ->  -. 
 A. x ( x  e.  y  ->  -.  A. z ( z  e.  x  ->  -.  z  e.  y ) ) )
 
Theoremaxinfprim 23223 ax-inf 7223 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
 |-  -.  A. x  -.  ( y  e.  z  ->  -.  (
 y  e.  x  ->  -.  A. y ( y  e.  x  ->  -.  A. z ( y  e.  z  ->  -.  z  e.  x ) ) ) )
 
Theoremaxacprim 23224 ax-ac 7969 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 26-Oct-2010.)
 |-  -.  A. x  -.  A. y A. z ( A. x  -.  ( y  e.  z  ->  -.  z  e.  w )  ->  -.  A. w  -.  A. y  -.  ( ( -.  A. w ( y  e.  z  ->  ( z  e.  w  ->  ( y  e.  w  ->  -.  w  e.  x ) ) )  ->  y  =  w )  ->  -.  ( y  =  w  ->  -.  A. w ( y  e.  z  ->  ( z  e.  w  ->  ( y  e.  w  ->  -.  w  e.  x ) ) ) ) ) )
 
16.6.2  Untangled classes
 
Theoremuntelirr 23225* We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 23316). 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.)
 |-  ( A. x  e.  A  -.  x  e.  x  ->  -.  A  e.  A )
 
Theoremuntuni 23226* The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.)
 |-  ( A. x  e.  U. A  -.  x  e.  x  <->  A. y  e.  A  A. x  e.  y  -.  x  e.  x )
 
Theoremuntsucf 23227* If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ y A   =>    |-  ( A. x  e.  A  -.  x  e.  x  ->  A. y  e. 
 suc  A  -.  y  e.  y )
 
Theoremunt0 23228 The null set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  A. x  e.  (/)  -.  x  e.  x
 
Theoremuntint 23229* If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
 |-  ( E. x  e.  A  A. y  e.  x  -.  y  e.  y  ->  A. y  e.  |^| A  -.  y  e.  y
 )
 
Theoremefrunt 23230* If  A is well-founded by  _E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
 |-  (  _E  Fr  A  ->  A. x  e.  A  -.  x  e.  x )
 
Theoremuntangtr 23231* A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.)
 |-  ( Tr  A  ->  ( A. x  e.  A  -.  x  e.  x  <->  A. x  e.  A  A. y  e.  x  -.  y  e.  y )
 )
 
16.6.3  Extra propositional calculus theorems
 
Theorem3orel1 23232 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)
 |-  ( -.  ph  ->  ( ( ph  \/  ps  \/  ch )  ->  ( ps  \/  ch ) ) )
 
Theorem3orel2 23233 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.)
 |-  ( -.  ps  ->  ( ( ph  \/  ps  \/  ch )  ->  ( ph  \/  ch ) ) )
 
Theorem3orel3 23234 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)
 |-  ( -.  ch  ->  ( ( ph  \/  ps  \/  ch )  ->  ( ph  \/  ps ) ) )
 
Theorem3pm3.2ni 23235 Triple negated disjuntion introduction. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  -.  ph   &    |-  -. 
 ps   &    |- 
 -.  ch   =>    |- 
 -.  ( ph  \/  ps 
 \/  ch )
 
Theorem3jaodd 23236 Double deduction form of 3jaoi 1250. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  et )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  et ) ) )   &    |-  ( ph  ->  ( ps  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ( ch 
 \/  th  \/  ta )  ->  et ) ) )
 
Theorem3orit 23237 Closed form of 3ori 1247, (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  (
 ( ph  \/  ps  \/  ch )  <->  ( ( -.  ph  /\  -.  ps )  ->  ch ) )
 
Theorem3mix1d 23238 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  \/  ch  \/  th ) )
 
Theorem3mix2d 23239 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  ps  \/  th ) )
 
Theorem3mix3d 23240 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  th  \/  ps ) )
 
Theorembiimpexp 23241 A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)
 |-  (
 ( ( ph  <->  ps )  ->  ch )  <->  ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  ch )
 ) )
 
Theorem3orel13 23242 Elimination of two disjuncts in a triple disjunction. (Contributed by Scott Fenton, 9-Jun-2011.)
 |-  (
 ( -.  ph  /\  -.  ch )  ->  ( ( ph  \/  ps  \/  ch )  ->  ps ) )
 
16.6.4  Misc. Useful Theorems
 
Theoremnepss 23243 Two classes are inequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)
 |-  ( A  =/=  B  <->  ( ( A  i^i  B )  C.  A  \/  ( A  i^i  B )  C.  B ) )
 
Theorem3ccased 23244 Triple disjunction form of ccased 918. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ph  ->  ( ( ch 
 /\  et )  ->  ps )
 )   &    |-  ( ph  ->  (
 ( ch  /\  ze )  ->  ps ) )   &    |-  ( ph  ->  ( ( ch 
 /\  si )  ->  ps )
 )   &    |-  ( ph  ->  (
 ( th  /\  et )  ->  ps ) )   &    |-  ( ph  ->  ( ( th  /\ 
 ze )  ->  ps )
 )   &    |-  ( ph  ->  (
 ( th  /\  si )  ->  ps ) )   &    |-  ( ph  ->  ( ( ta 
 /\  et )  ->  ps )
 )   &    |-  ( ph  ->  (
 ( ta  /\  ze )  ->  ps ) )   &    |-  ( ph  ->  ( ( ta 
 /\  si )  ->  ps )
 )   =>    |-  ( ph  ->  (
 ( ( ch  \/  th 
 \/  ta )  /\  ( et  \/  ze  \/  si ) )  ->  ps )
 )
 
Theoremdfso3 23245* Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.)
 |-  ( R  Or  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z )  /\  ( x R y  \/  x  =  y  \/  y R x ) ) )
 
Theorembrtpid1 23246 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { <. A ,  B >. ,  C ,  D } B
 
Theorembrtpid2 23247 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { C ,  <. A ,  B >. ,  D } B
 
Theorembrtpid3 23248 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { C ,  D ,  <. A ,  B >. } B
 
16.6.5  Properties of reals and complexes
 
Theoremsqdivzi 23249 Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  (
 ( A  /  B ) ^ 2 )  =  ( ( A ^
 2 )  /  ( B ^ 2 ) ) )
 
Theoremdivelunit 23250 A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  e.  ( 0 [,] 1 )  <->  A  <_  B ) )
 
Theorempm2.61iine 23251 Equality version of pm2.61ii 159. (Contributed by Scott Fenton, 13-Jun-2013.)
 |-  (
 ( A  =/=  C  /\  B  =/=  D ) 
 ->  ph )   &    |-  ( A  =  C  ->  ph )   &    |-  ( B  =  D  ->  ph )   =>    |-  ph
 
Theoremdedekind 23252* The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 8695 with appropriate adjustments, states that, if  A completely preceeds  B, then there is some number separating the two of them. (Contributed by Scott Fenton, 13-Jun-2013.)
 |-  (
 ( A  C_  RR  /\  B  C_  RR  /\  A. x  e.  A  A. y  e.  B  x  <  y
 )  ->  E. z  e.  RR  A. x  e.  A  A. y  e.  B  ( x  <_  z  /\  z  <_  y
 ) )
 
Theoremdedekindle 23253* The Dedekind cut theorem, with the hypothesis weakened to only require non-strict less than. (Contributed by Scott Fenton, 2-Jul-2013.)
 |-  (
 ( A  C_  RR  /\  B  C_  RR  /\  A. x  e.  A  A. y  e.  B  x  <_  y
 )  ->  E. z  e.  RR  A. x  e.  A  A. y  e.  B  ( x  <_  z  /\  z  <_  y
 ) )
 
Theoremmulcan1g 23254 A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  =  ( A  x.  C )  <->  ( A  =  0  \/  B  =  C ) ) )
 
Theoremmulcan2g 23255 A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  C )  =  ( B  x.  C )  <->  ( A  =  B  \/  C  =  0 ) ) )
 
Theoremmulge0b 23256 A condition for multiplication to be non-negative. (Contributed by Scott Fenton, 25-Jun-2013.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  ( A  x.  B )  <->  ( ( A 
 <_  0  /\  B  <_  0 )  \/  ( 0 
 <_  A  /\  0  <_  B ) ) ) )
 
Theoremmulle0b 23257 A condition for multiplication to be non-positive. (Contributed by Scott Fenton, 25-Jun-2013.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  <_  0  <->  ( ( A 
 <_  0  /\  0  <_  B )  \/  (
 0  <_  A  /\  B  <_  0 ) ) ) )
 
Theoremmulsuble0b 23258 A condition for multiplication of subtraction to be non-positive. (Contributed by Scott Fenton, 25-Jun-2013.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( ( A  -  B )  x.  ( C  -  B ) )  <_  0  <->  ( ( A 
 <_  B  /\  B  <_  C )  \/  ( C 
 <_  B  /\  B  <_  A ) ) ) )
 
Theoremrelin01 23259 An interval law for less than or equal. (Contributed by Scott Fenton, 27-Jun-2013.)
 |-  ( A  e.  RR  ->  ( A  <_  0  \/  ( 0  <_  A  /\  A  <_  1 )  \/  1  <_  A ) )
 
Theoremsubdivcomb1 23260 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( (
 ( C  x.  A )  -  B )  /  C )  =  ( A  -  ( B  /  C ) ) )
 
Theoremsubdivcomb2 23261 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  -  ( C  x.  B ) )  /  C )  =  (
 ( A  /  C )  -  B ) )
 
Theoremsubeqrev 23262 Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  -  B )  =  ( C  -  D )  <->  ( B  -  A )  =  ( D  -  C ) ) )
 
Theoremfznatpl1 23263 Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)
 |-  (
 ( N  e.  NN  /\  I  e.  ( 1
 ... ( N  -  1 ) ) ) 
 ->  ( I  +  1 )  e.  ( 1
 ... N ) )
 
Theoremsupfz 23264 The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  sup (
 ( M ... N ) ,  ZZ ,  <  )  =  N )
 
Theoreminffz 23265 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  sup (
 ( M ... N ) ,  ZZ ,  `'  <  )  =  M )
 
Theorembcnm1 23266 The binomial coefficent of  ( N  -  1 ) is  N. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  ( N  -  1 ) )  =  N )
 
Theoremfz0n 23267 The sequence  ( 0 ... ( N  -  1 ) ) is empty iff  N is zero. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  NN0  ->  (
 ( 0 ... ( N  -  1 ) )  =  (/)  <->  N  =  0
 ) )
 
Theorem4bc3eq4 23268 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
 |-  (
 4  _C  3 )  =  4
 
Theorem4bc2eq6 23269 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  (
 4  _C  2 )  =  6
 
Theoremhalfthird 23270 Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  (
 ( 1  /  2
 )  -  ( 1 
 /  3 ) )  =  ( 1  / 
 6 )
 
Theorem5recm6rec 23271 One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  (
 ( 1  /  5
 )  -  ( 1 
 /  6 ) )  =  ( 1  / ; 3 0 )
 
16.6.6  Greatest common divisor and divisibility
 
Theorempdivsq 23272 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( P  e.  Prime  /\  M  e.  ZZ )  ->  ( P  ||  M  <->  P 
 ||  ( M ^
 2 ) ) )
 
Theoremdvdspw 23273 Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  ->  ( K  ||  M  ->  K  ||  ( M ^ N ) ) )
 
Theoremgcd32 23274 Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  C )  =  ( ( A 
 gcd  C )  gcd  B ) )
 
Theoremgcdabsorb 23275 Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  B )  =  ( A  gcd  B ) )
 
16.6.7  Properties of relationships
 
Theorembrtp 23276 A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  X  e.  _V   &    |-  Y  e.  _V   =>    |-  ( X { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } Y  <->  ( ( X  =  A  /\  Y  =  B )  \/  ( X  =  C  /\  Y  =  D )  \/  ( X  =  E  /\  Y  =  F ) ) )
 
Theoremdftr6 23277 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
 |-  A  e.  _V   =>    |-  ( Tr  A  <->  A  e.  ( _V  \  ran  ( (  _E  o.  _E  )  \  _E  ) ) )
 
Theoremcoep 23278* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A (  _E  o.  R ) B  <->  E. x  e.  B  A R x )
 
Theoremcoepr 23279* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A ( R  o.  `'  _E  ) B  <->  E. x  e.  A  x R B )
 
Theoremdffr5 23280 A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)
 |-  ( R  Fr  A  <->  ( ~P A  \  { (/) } )  C_  ran  (  _E  \  (  _E  o.  `' R ) ) )
 
Theoremdfso2 23281 Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)
 |-  ( R  Or  A  <->  ( R  Po  A  /\  ( A  X.  A )  C_  ( R  u.  (  _I  u.  `' R ) ) ) )
 
Theoremdfpo2 23282 Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.)
 |-  ( R  Po  A  <->  ( ( R  i^i  (  _I  |`  A ) )  =  (/)  /\  (
 ( R  i^i  ( A  X.  A ) )  o.  ( R  i^i  ( A  X.  A ) ) )  C_  R ) )
 
Theorembr8 23283* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  (
 e  =  E  ->  ( ta  <->  et ) )   &    |-  (
 f  =  F  ->  ( et  <->  ze ) )   &    |-  (
 g  =  G  ->  ( ze  <->  si ) )   &    |-  ( h  =  H  ->  (
 si 
 <->  rh ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  E. g  e.  P  E. h  e.  P  ( p  = 
 <. <. a ,  b >. ,  <. c ,  d >.
 >.  /\  q  =  <. <.
 e ,  f >. , 
 <. g ,  h >. >.  /\  ph ) }   =>    |-  ( ( ( X  e.  S  /\  A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q  /\  E  e.  Q )  /\  ( F  e.  Q  /\  G  e.  Q  /\  H  e.  Q )
 )  ->  ( <. <. A ,  B >. , 
 <. C ,  D >. >. R <. <. E ,  F >. ,  <. G ,  H >.
 >. 
 <->  rh ) )
 
Theorembr6 23284* Substitution for an six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  (
 e  =  E  ->  ( ta  <->  et ) )   &    |-  (
 f  =  F  ->  ( et  <->  ze ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  ( p  = 
 <. a ,  <. b ,  c >. >.  /\  q  =  <. d ,  <. e ,  f >. >.  /\  ph ) }   =>    |-  (
 ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q  /\  C  e.  Q )  /\  ( D  e.  Q  /\  E  e.  Q  /\  F  e.  Q )
 )  ->  ( <. A ,  <. B ,  C >.
 >. R <. D ,  <. E ,  F >. >.  <->  ze ) )
 
Theorembr4 23285* Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( p  = 
 <. a ,  b >.  /\  q  =  <. c ,  d >.  /\  ph ) }   =>    |-  ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q ) )  ->  ( <. A ,  B >. R <. C ,  D >.  <->  ta ) )
 
Theoremdfres3 23286 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  ran  A )
 )
 
Theoremcnvco1 23287 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( `' A  o.  B )  =  ( `' B  o.  A )
 
Theoremcnvco2 23288 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( A  o.  `' B )  =  ( B  o.  `' A )
 
16.6.8  Properties of functions and mappings
 
Theoremfunpsstri 23289 A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
 |-  (
 ( Fun  H  /\  ( F  C_  H  /\  G  C_  H )  /\  ( dom  F  C_  dom  G  \/  dom  G  C_  dom  F ) )  ->  ( F 
 C.  G  \/  F  =  G  \/  G  C.  F ) )
 
Theoremfundmpss 23290 If a class  F is a proper subset of a function  G, then  dom  F  C.  dom  G. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  ( Fun  G  ->  ( F  C.  G  ->  dom  F  C.  dom 
 G ) )
 
Theoremfvresval 23291 The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( ( F  |`  B ) `
  A )  =  ( F `  A )  \/  ( ( F  |`  B ) `  A )  =  (/) )
 
Theoremmptrel 23292 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Rel  ( x  e.  A  |->  B )
 
Theoremfunsseq 23293 Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  (
 ( Fun  F  /\  Fun 
 G  /\  dom  F  =  dom  G )  ->  ( F  =  G  <->  F  C_  G ) )
 
Theoremfununiq 23294 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( Fun  F  ->  ( ( A F B  /\  A F C ) 
 ->  B  =  C ) )
 
Theoremfunbreq 23295 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( ( Fun  F  /\  A F B ) 
 ->  ( A F C  <->  B  =  C ) )
 
Theoremmpteq12d 23296 An equality inference for the maps to notation. Compare mpteq12dv 3995. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ x ph   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
 
Theoremfprb 23297* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  =/=  B  ->  ( F : { A ,  B } --> R  <->  E. x  e.  R  E. y  e.  R  F  =  { <. A ,  x >. ,  <. B ,  y >. } ) )
 
Theorembr1steq 23298 Uniqueness condition for binary relationship over the  1st relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >. 1st C  <->  C  =  A )
 
Theorembr2ndeq 23299 Uniqueness condition for binary relationship over the  2nd relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >. 2nd C  <->  C  =  B )
 
Theoremdfdm5 23300 Definition of domain in terms of 
1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  dom  A  =  ( ( 1st  |`  ( _V  X.  _V ) ) " A )
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