HomeHome Intuitionistic Logic Explorer
Theorem List (p. 62 of 102)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnawordex 6101* Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  C_  B 
 <-> 
 E. x  e.  om  ( A  +o  x )  =  B )
 )
 
Theoremnnm00 6102 The product of two natural numbers is zero iff at least one of them is zero. (Contributed by Jim Kingdon, 11-Nov-2004.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( ( A  .o  B )  =  (/) 
 <->  ( A  =  (/)  \/  B  =  (/) ) ) )
 
2.6.24  Equivalence relations and classes
 
Syntaxwer 6103 Extend the definition of a wff to include the equivalence predicate.
 wff  R  Er  A
 
Syntaxcec 6104 Extend the definition of a class to include equivalence class.
 class  [ A ] R
 
Syntaxcqs 6105 Extend the definition of a class to include quotient set.
 class  ( A /. R )
 
Definitiondf-er 6106 Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 6107 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6126, ersymb 6120, and ertr 6121. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)
 |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
 C_  R ) )
 
Theoremdfer2 6107* Alternate definition of equivalence predicate. (Contributed by NM, 3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  A. x A. y A. z
 ( ( x R y  ->  y R x )  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) ) )
 
Definitiondf-ec 6108 Define the  R-coset of  A. Exercise 35 of [Enderton] p. 61. This is called the equivalence class of  A modulo  R when  R is an equivalence relation (i.e. when  Er  R; see dfer2 6107). In this case,  A is a representative (member) of the equivalence class  [ A ] R, which contains all sets that are equivalent to  A. Definition of [Enderton] p. 57 uses the notation  [ A ] (subscript)  R, although we simply follow the brackets by  R since we don't have subscripted expressions. For an alternate definition, see dfec2 6109. (Contributed by NM, 23-Jul-1995.)
 |- 
 [ A ] R  =  ( R " { A } )
 
Theoremdfec2 6109* Alternate definition of  R-coset of  A. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  V  ->  [ A ] R  =  { y  |  A R y } )
 
Theoremecexg 6110 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
 |-  ( R  e.  B  ->  [ A ] R  e.  _V )
 
Theoremecexr 6111 An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  [ B ] R  ->  B  e.  _V )
 
Definitiondf-qs 6112* Define quotient set.  R is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)
 |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
 
Theoremereq1 6113 Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  =  S  ->  ( R  Er  A  <->  S  Er  A ) )
 
Theoremereq2 6114 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( A  =  B  ->  ( R  Er  A  <->  R  Er  B ) )
 
Theoremerrel 6115 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  Rel  R )
 
Theoremerdm 6116 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  dom  R  =  A )
 
Theoremercl 6117 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  A  e.  X )
 
Theoremersym 6118 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  B R A )
 
Theoremercl2 6119 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  B  e.  X )
 
Theoremersymb 6120 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   =>    |-  ( ph  ->  ( A R B  <->  B R A ) )
 
Theoremertr 6121 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   =>    |-  ( ph  ->  (
 ( A R B  /\  B R C ) 
 ->  A R C ) )
 
Theoremertrd 6122 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremertr2d 6123 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  C R A )
 
Theoremertr3d 6124 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  B R A )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremertr4d 6125 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   &    |-  ( ph  ->  C R B )   =>    |-  ( ph  ->  A R C )
 
Theoremerref 6126 An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  A R A )
 
Theoremercnv 6127 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  `' R  =  R )
 
Theoremerrn 6128 The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  ran  R  =  A )
 
Theoremerssxp 6129 An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  R  C_  ( A  X.  A ) )
 
Theoremerex 6130 An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  ( A  e.  V  ->  R  e.  _V )
 )
 
Theoremerexb 6131 An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  ( R  e.  _V  <->  A  e.  _V ) )
 
Theoremiserd 6132* A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ( ph  /\  x R y )  ->  y R x )   &    |-  (
 ( ph  /\  ( x R y  /\  y R z ) ) 
 ->  x R z )   &    |-  ( ph  ->  ( x  e.  A  <->  x R x ) )   =>    |-  ( ph  ->  R  Er  A )
 
Theorembrdifun 6133 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B 
 <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
 
Theoremswoer 6134* Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   =>    |-  ( ph  ->  R  Er  X )
 
Theoremswoord1 6135* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  ( A  .<  C  <->  B  .<  C ) )
 
Theoremswoord2 6136* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  ( C  .<  A  <->  C  .<  B ) )
 
Theoremeqerlem 6137* Lemma for eqer 6138. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  ( x  =  y 
 ->  A  =  B )   &    |-  R  =  { <. x ,  y >.  |  A  =  B }   =>    |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
 
Theoremeqer 6138* Equivalence relation involving equality of dependent classes  A
( x ) and  B ( y ). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( x  =  y 
 ->  A  =  B )   &    |-  R  =  { <. x ,  y >.  |  A  =  B }   =>    |-  R  Er  _V
 
Theoremider 6139 The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
 |- 
 _I  Er  _V
 
Theorem0er 6140 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
 |-  (/)  Er  (/)
 
Theoremeceq1 6141 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )
 
Theoremeceq1d 6142 Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  [ A ] C  =  [ B ] C )
 
Theoremeceq2 6143 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  [ C ] A  =  [ C ] B )
 
Theoremelecg 6144 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  [ B ] R  <->  B R A ) )
 
Theoremelec 6145 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  e.  [ B ] R  <->  B R A )
 
Theoremrelelec 6146 Membership in an equivalence class when  R is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( Rel  R  ->  ( A  e.  [ B ] R  <->  B R A ) )
 
Theoremecss 6147 An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   =>    |-  ( ph  ->  [ A ] R  C_  X )
 
Theoremecdmn0m 6148* A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)
 |-  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R )
 
Theoremereldm 6149 Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  [ A ] R  =  [ B ] R )   =>    |-  ( ph  ->  ( A  e.  X  <->  B  e.  X ) )
 
Theoremerth 6150 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R ) )
 
Theoremerth2 6151 Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  B  e.  X )   =>    |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R ) )
 
Theoremerthi 6152 Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  [ A ] R  =  [ B ] R )
 
Theoremecidsn 6153 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
 |- 
 [ A ]  _I  =  { A }
 
Theoremqseq1 6154 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  ( A /. C )  =  ( B /. C ) )
 
Theoremqseq2 6155 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  ( C /. A )  =  ( C /. B ) )
 
Theoremelqsg 6156* Closed form of elqs 6157. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( B  e.  V  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R ) )
 
Theoremelqs 6157* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  B  e.  _V   =>    |-  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R )
 
Theoremelqsi 6158* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( B  e.  ( A /. R )  ->  E. x  e.  A  B  =  [ x ] R )
 
Theoremecelqsg 6159 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( R  e.  V  /\  B  e.  A )  ->  [ B ] R  e.  ( A /. R ) )
 
Theoremecelqsi 6160 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  R  e.  _V   =>    |-  ( B  e.  A  ->  [ B ] R  e.  ( A /. R ) )
 
Theoremecopqsi 6161 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)
 |-  R  e.  _V   &    |-  S  =  ( ( A  X.  A ) /. R )   =>    |-  ( ( B  e.  A  /\  C  e.  A )  ->  [ <. B ,  C >. ] R  e.  S )
 
Theoremqsexg 6162 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  V  ->  ( A /. R )  e.  _V )
 
Theoremqsex 6163 A quotient set exists. (Contributed by NM, 14-Aug-1995.)
 |-  A  e.  _V   =>    |-  ( A /. R )  e.  _V
 
Theoremuniqs 6164 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
 |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A ) )
 
Theoremqsss 6165 A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  A )   =>    |-  ( ph  ->  ( A /. R )  C_  ~P A )
 
Theoremuniqs2 6166 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  ( ph  ->  R  Er  A )   &    |-  ( ph  ->  R  e.  V )   =>    |-  ( ph  ->  U. ( A /. R )  =  A )
 
Theoremsnec 6167 The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  A  e.  _V   =>    |-  { [ A ] R }  =  ( { A } /. R )
 
Theoremecqs 6168 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
 |-  R  e.  _V   =>    |-  [ A ] R  =  U. ( { A } /. R )
 
Theoremecid 6169 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  A  e.  _V   =>    |-  [ A ] `'  _E  =  A
 
Theoremecidg 6170 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.)
 |-  ( A  e.  V  ->  [ A ] `'  _E  =  A )
 
Theoremqsid 6171 A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( A /. `'  _E  )  =  A
 
Theoremectocld 6172* Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  S  =  ( B
 /. R )   &    |-  ( [ x ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 ( ch  /\  x  e.  B )  ->  ph )   =>    |-  (
 ( ch  /\  A  e.  S )  ->  ps )
 
Theoremectocl 6173* Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  S  =  ( B
 /. R )   &    |-  ( [ x ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  e.  B  ->  ph )   =>    |-  ( A  e.  S  ->  ps )
 
Theoremelqsn0m 6174* An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)
 |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  E. x  x  e.  B )
 
Theoremelqsn0 6175 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
 |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  B  =/=  (/) )
 
Theoremecelqsdm 6176 Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)
 |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) ) 
 ->  B  e.  A )
 
Theoremxpiderm 6177* A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by Jim Kingdon, 22-Aug-2019.)
 |-  ( E. x  x  e.  A  ->  ( A  X.  A )  Er  A )
 
Theoremiinerm 6178* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  R  Er  B )  ->  |^|_ x  e.  A  R  Er  B )
 
Theoremriinerm 6179* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  R  Er  B )  ->  (
 ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B )
 
Theoremerinxp 6180 A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  A )   &    |-  ( ph  ->  B 
 C_  A )   =>    |-  ( ph  ->  ( R  i^i  ( B  X.  B ) )  Er  B )
 
Theoremecinxp 6181 Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  ( ( ( R
 " A )  C_  A  /\  B  e.  A )  ->  [ B ] R  =  [ B ] ( R  i^i  ( A  X.  A ) ) )
 
Theoremqsinxp 6182 Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ( R " A )  C_  A  ->  ( A /. R )  =  ( A /. ( R  i^i  ( A  X.  A ) ) ) )
 
Theoremqsel 6183 If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( ( R  Er  X  /\  B  e.  ( A /. R )  /\  C  e.  B )  ->  B  =  [ C ] R )
 
Theoremqliftlem 6184*  F, a function lift, is a subset of  R  X.  S. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ( ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
 
Theoremqliftrel 6185*  F, a function lift, is a subset of  R  X.  S. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ph  ->  F  C_  ( ( X /. R )  X.  Y ) )
 
Theoremqliftel 6186* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ph  ->  ( [ C ] R F D 
 <-> 
 E. x  e.  X  ( C R x  /\  D  =  A )
 ) )
 
Theoremqliftel1 6187* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ( ph  /\  x  e.  X )  ->  [ x ] R F A )
 
Theoremqliftfun 6188* The function  F is the unique function defined by  F `  [
x ]  =  A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   &    |-  ( x  =  y 
 ->  A  =  B )   =>    |-  ( ph  ->  ( Fun  F  <->  A. x A. y ( x R y  ->  A  =  B )
 ) )
 
Theoremqliftfund 6189* The function  F is the unique function defined by  F `  [
x ]  =  A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   &    |-  ( x  =  y 
 ->  A  =  B )   &    |-  ( ( ph  /\  x R y )  ->  A  =  B )   =>    |-  ( ph  ->  Fun  F )
 
Theoremqliftfuns 6190* The function  F is the unique function defined by  F `  [
x ]  =  A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ph  ->  ( Fun  F  <->  A. y A. z
 ( y R z 
 ->  [_ y  /  x ]_ A  =  [_ z  /  x ]_ A ) ) )
 
Theoremqliftf 6191* The domain and range of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ph  ->  ( Fun  F  <->  F : ( X
 /. R ) --> Y ) )
 
Theoremqliftval 6192* The value of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   &    |-  ( x  =  C  ->  A  =  B )   &    |-  ( ph  ->  Fun  F )   =>    |-  ( ( ph  /\  C  e.  X )  ->  ( F `  [ C ] R )  =  B )
 
Theoremecoptocl 6193* Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)
 |-  S  =  ( ( B  X.  C )
 /. R )   &    |-  ( [ <. x ,  y >. ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 ( x  e.  B  /\  y  e.  C )  ->  ph )   =>    |-  ( A  e.  S  ->  ps )
 
Theorem2ecoptocl 6194* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)
 |-  S  =  ( ( C  X.  D )
 /. R )   &    |-  ( [ <. x ,  y >. ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  ( [ <. z ,  w >. ] R  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 ( ( x  e.  C  /\  y  e.  D )  /\  (
 z  e.  C  /\  w  e.  D )
 )  ->  ph )   =>    |-  ( ( A  e.  S  /\  B  e.  S )  ->  ch )
 
Theorem3ecoptocl 6195* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)
 |-  S  =  ( ( D  X.  D )
 /. R )   &    |-  ( [ <. x ,  y >. ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  ( [ <. z ,  w >. ] R  =  B  ->  ( ps  <->  ch ) )   &    |-  ( [ <. v ,  u >. ] R  =  C  ->  ( ch  <->  th ) )   &    |-  (
 ( ( x  e.  D  /\  y  e.  D )  /\  (
 z  e.  D  /\  w  e.  D )  /\  ( v  e.  D  /\  u  e.  D ) )  ->  ph )   =>    |-  (
 ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  th )
 
Theorembrecop 6196* Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)
 |- 
 .~  e.  _V   &    |-  .~  Er  ( G  X.  G )   &    |-  H  =  ( ( G  X.  G ) /.  .~  )   &    |- 
 .<_  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ] 
 .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) }   &    |-  (
 ( ( ( z  e.  G  /\  w  e.  G )  /\  ( A  e.  G  /\  B  e.  G )
 )  /\  ( (
 v  e.  G  /\  u  e.  G )  /\  ( C  e.  G  /\  D  e.  G ) ) )  ->  (
 ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ] 
 .~  =  [ <. C ,  D >. ]  .~  )  ->  ( ph  <->  ps ) ) )   =>    |-  ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
 )  ->  ( [ <. A ,  B >. ] 
 .~  .<_  [ <. C ,  D >. ]  .~  <->  ps ) )
 
Theoremeroveu 6197* Lemma for eroprf 6199. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  J  =  ( A
 /. R )   &    |-  K  =  ( B /. S )   &    |-  ( ph  ->  T  e.  Z )   &    |-  ( ph  ->  R  Er  U )   &    |-  ( ph  ->  S  Er  V )   &    |-  ( ph  ->  T  Er  W )   &    |-  ( ph  ->  A 
 C_  U )   &    |-  ( ph  ->  B  C_  V )   &    |-  ( ph  ->  C  C_  W )   &    |-  ( ph  ->  .+ 
 : ( A  X.  B ) --> C )   &    |-  ( ( ph  /\  (
 ( r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B ) ) ) 
 ->  ( ( r R s  /\  t S u )  ->  (
 r  .+  t ) T ( s  .+  u ) ) )   =>    |-  ( ( ph  /\  ( X  e.  J  /\  Y  e.  K )
 )  ->  E! z E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T ) )
 
Theoremerovlem 6198* Lemma for eroprf 6199. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  J  =  ( A
 /. R )   &    |-  K  =  ( B /. S )   &    |-  ( ph  ->  T  e.  Z )   &    |-  ( ph  ->  R  Er  U )   &    |-  ( ph  ->  S  Er  V )   &    |-  ( ph  ->  T  Er  W )   &    |-  ( ph  ->  A 
 C_  U )   &    |-  ( ph  ->  B  C_  V )   &    |-  ( ph  ->  C  C_  W )   &    |-  ( ph  ->  .+ 
 : ( A  X.  B ) --> C )   &    |-  ( ( ph  /\  (
 ( r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B ) ) ) 
 ->  ( ( r R s  /\  t S u )  ->  (
 r  .+  t ) T ( s  .+  u ) ) )   &    |-  .+^ 
 =  { <. <. x ,  y >. ,  z >.  | 
 E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
 q ] S ) 
 /\  z  =  [
 ( p  .+  q
 ) ] T ) }   =>    |-  ( ph  ->  .+^  =  ( x  e.  J ,  y  e.  K  |->  ( iota
 z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
 q ] S ) 
 /\  z  =  [
 ( p  .+  q
 ) ] T ) ) ) )
 
Theoremeroprf 6199* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  J  =  ( A
 /. R )   &    |-  K  =  ( B /. S )   &    |-  ( ph  ->  T  e.  Z )   &    |-  ( ph  ->  R  Er  U )   &    |-  ( ph  ->  S  Er  V )   &    |-  ( ph  ->  T  Er  W )   &    |-  ( ph  ->  A 
 C_  U )   &    |-  ( ph  ->  B  C_  V )   &    |-  ( ph  ->  C  C_  W )   &    |-  ( ph  ->  .+ 
 : ( A  X.  B ) --> C )   &    |-  ( ( ph  /\  (
 ( r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B ) ) ) 
 ->  ( ( r R s  /\  t S u )  ->  (
 r  .+  t ) T ( s  .+  u ) ) )   &    |-  .+^ 
 =  { <. <. x ,  y >. ,  z >.  | 
 E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
 q ] S ) 
 /\  z  =  [
 ( p  .+  q
 ) ] T ) }   &    |-  ( ph  ->  R  e.  X )   &    |-  ( ph  ->  S  e.  Y )   &    |-  L  =  ( C
 /. T )   =>    |-  ( ph  ->  .+^  : ( J  X.  K )
 --> L )
 
Theoremeroprf2 6200* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  J  =  ( A
 /.  .~  )   &    |-  .+^  =  { <.
 <. x ,  y >. ,  z >.  |  E. p  e.  A  E. q  e.  A  ( ( x  =  [ p ]  .~  /\  y  =  [
 q ]  .~  )  /\  z  =  [
 ( p  .+  q
 ) ]  .~  ) }   &    |-  ( ph  ->  .~  e.  X )   &    |-  ( ph  ->  .~ 
 Er  U )   &    |-  ( ph  ->  A  C_  U )   &    |-  ( ph  ->  .+  :
 ( A  X.  A )
 --> A )   &    |-  ( ( ph  /\  ( ( r  e.  A  /\  s  e.  A )  /\  (
 t  e.  A  /\  u  e.  A )
 ) )  ->  (
 ( r  .~  s  /\  t  .~  u ) 
 ->  ( r  .+  t
 )  .~  ( s  .+  u ) ) )   =>    |-  ( ph  ->  .+^  : ( J  X.  J ) --> J )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10124
  Copyright terms: Public domain < Previous  Next >