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Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrelexpsucl 23401 A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  ( R ^ r ( N  +  1 ) )  =  ( R  o.  ( R ^ r N ) ) ) )
 
Theoremrelexpcnv 23402 Distributivity of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  `' ( R ^ r N )  =  ( `' R ^ r N ) ) )
 
TheoremrelexpexOLD 23403 Obsolete; use ovex 5817 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 23404 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 23405 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 23406 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 23407 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 23408 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 23409* 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 23410* 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 23411 Extend class notation with recursively defined reflexive, transitive closure.
 class  t *rec
 
Definitiondf-rtrclrec 23412* 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 23413* 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 23414 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 23415 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 23416 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 23417* 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 23418 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 23419* 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 ) )
 
18.7  Mathbox for Scott Fenton
 
18.7.1  ZFC Axioms in primitive form
 
Theoremaxextprim 23420 ax-ext 2239 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 23421 ax-rep 4105 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 23422 ax-un 4484 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 23423 ax-pow 4160 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 23424 ax-reg 7274 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 23425 ax-inf 7307 without distinct variable conditions or defined symbols. (New usage is discouraged.) (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 23426 ax-ac 8053 without distinct variable conditions or defined symbols. (New usage is discouraged.) (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 ) ) ) ) ) )
 
18.7.2  Untangled classes
 
Theoremuntelirr 23427* We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 23518). 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 23428* 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 23429* 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 23430 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 23431* 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 23432* 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 23433* 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 )
 )
 
18.7.3  Extra propositional calculus theorems
 
Theorem3orel1 23434 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 23435 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 23436 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 23437 Triple negated disjuntion introduction. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  -.  ph   &    |-  -. 
 ps   &    |- 
 -.  ch   =>    |- 
 -.  ( ph  \/  ps 
 \/  ch )
 
Theorem3jaodd 23438 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 23439 Closed form of 3ori 1247, (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  (
 ( ph  \/  ps  \/  ch )  <->  ( ( -.  ph  /\  -.  ps )  ->  ch ) )
 
Theorem3mix1d 23440 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  \/  ch  \/  th ) )
 
Theorem3mix2d 23441 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  ps  \/  th ) )
 
Theorem3mix3d 23442 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  th  \/  ps ) )
 
Theorembiimpexp 23443 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 23444 Elimination of two disjuncts in a triple disjunction. (Contributed by Scott Fenton, 9-Jun-2011.)
 |-  (
 ( -.  ph  /\  -.  ch )  ->  ( ( ph  \/  ps  \/  ch )  ->  ps ) )
 
18.7.4  Misc. Useful Theorems
 
Theoremnepss 23445 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 23446 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 23447* 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 23448 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { <. A ,  B >. ,  C ,  D } B
 
Theorembrtpid2 23449 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { C ,  <. A ,  B >. ,  D } B
 
Theorembrtpid3 23450 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { C ,  D ,  <. A ,  B >. } B
 
18.7.5  Properties of reals and complexes
 
Theoremsqdivzi 23451 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 23452 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 23453 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 23454* The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 8783 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 23455* 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 23456 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 23457 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 23458 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 23459 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 23460 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 23461 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 23462 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 23463 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 23464 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 23465 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 23466 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 23467 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 23468 The binomial coefficent of  ( N  -  1 ) is  N. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  ( N  -  1 ) )  =  N )
 
Theoremfz0n 23469 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 23470 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
 |-  (
 4  _C  3 )  =  4
 
Theorem4bc2eq6 23471 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  (
 4  _C  2 )  =  6
 
Theoremhalfthird 23472 Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  (
 ( 1  /  2
 )  -  ( 1 
 /  3 ) )  =  ( 1  / 
 6 )
 
Theorem5recm6rec 23473 One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  (
 ( 1  /  5
 )  -  ( 1 
 /  6 ) )  =  ( 1  / ; 3 0 )
 
18.7.6  Greatest common divisor and divisibility
 
Theorempdivsq 23474 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 23475 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 23476 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 23477 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 ) )
 
18.7.7  Properties of relationships
 
Theorembrtp 23478 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 23479 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 23480* 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 23481* 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 23482 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 23483 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 23484 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 23485* 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 23486* 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 23487* 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 23488 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 23489 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( `' A  o.  B )  =  ( `' B  o.  A )
 
Theoremcnvco2 23490 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( A  o.  `' B )  =  ( B  o.  `' A )
 
18.7.8  Properties of functions and mappings
 
Theoremfunpsstri 23491 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 23492 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 23493 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 23494 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Rel  ( x  e.  A  |->  B )
 
Theoremfunsseq 23495 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 23496 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 23497 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 23498 An equality inference for the maps to notation. Compare mpteq12dv 4072. (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 23499* 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 23500 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 )
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