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Theorem List for Metamath Proof Explorer - 26001-26100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremidlsubcl 26001 An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R )
 )  /\  ( A  e.  I  /\  B  e.  I ) )  ->  ( A D B )  e.  I )
 
Theoremrngoidl 26002 A ring  R is an  R ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R )
 )
 
Theorem0idl 26003 The set containing only  0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R ) )
 
Theorem1idl 26004 Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  ->  ( U  e.  I  <->  I  =  X ) )
 
Theorem0rngo 26005 In a ring,  0  =  1 iff the ring contains only  0. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( R  e.  RingOps  ->  ( Z  =  U  <->  X  =  { Z }
 ) )
 
Theoremdivrngidl 26006 The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  DivRingOps  ->  ( Idl `  R )  =  { { Z } ,  X } )
 
Theoremintidl 26007 The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  (
 ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) ) 
 ->  |^| C  e.  ( Idl `  R ) )
 
Theoreminidl 26008 The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R ) )  ->  ( I  i^i  J )  e.  ( Idl `  R ) )
 
Theoremunichnidl 26009* The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  (
 ( R  e.  RingOps  /\  ( C  =/=  (/)  /\  C  C_  ( Idl `  R )  /\  A. i  e.  C  A. j  e.  C  ( i  C_  j  \/  j  C_  i
 ) ) )  ->  U. C  e.  ( Idl `  R ) )
 
Theoremkeridl 26010 The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)
 |-  G  =  ( 1st `  S )   &    |-  Z  =  (GId `  G )   =>    |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( `' F " { Z }
 )  e.  ( Idl `  R ) )
 
Theorempridlval 26011* The class of prime ideals of a ring 
R. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  ( PrIdl `  R )  =  { i  e.  ( Idl `  R )  |  ( i  =/=  X  /\  A. a  e.  ( Idl `  R ) A. b  e.  ( Idl `  R ) ( A. x  e.  a  A. y  e.  b  ( x H y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) } )
 
Theoremispridl 26012* The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  ( P  e.  ( PrIdl `  R )  <->  ( P  e.  ( Idl `  R )  /\  P  =/=  X  /\  A. a  e.  ( Idl `  R ) A. b  e.  ( Idl `  R ) ( A. x  e.  a  A. y  e.  b  ( x H y )  e.  P  ->  ( a  C_  P  \/  b  C_  P ) ) ) ) )
 
Theorempridlidl 26013 A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  (
 ( R  e.  RingOps  /\  P  e.  ( PrIdl `  R ) )  ->  P  e.  ( Idl `  R ) )
 
Theorempridlnr 26014 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  P  e.  ( PrIdl `  R ) )  ->  P  =/=  X )
 
Theorempridl 26015* The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  H  =  ( 2nd `  R )   =>    |-  ( ( ( R  e.  RingOps  /\  P  e.  ( PrIdl `  R )
 )  /\  ( A  e.  ( Idl `  R )  /\  B  e.  ( Idl `  R )  /\  A. x  e.  A  A. y  e.  B  ( x H y )  e.  P ) )  ->  ( A  C_  P  \/  B  C_  P ) )
 
Theoremispridl2 26016* A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 26048 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  ( P  e.  ( Idl `  R )  /\  P  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P ) ) ) )  ->  P  e.  ( PrIdl `  R )
 )
 
Theoremmaxidlval 26017* The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  ( MaxIdl `  R )  =  {
 i  e.  ( Idl `  R )  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
 C_  j  ->  (
 j  =  i  \/  j  =  X ) ) ) } )
 
Theoremismaxidl 26018* The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  ( M  e.  ( MaxIdl `  R ) 
 <->  ( M  e.  ( Idl `  R )  /\  M  =/=  X  /\  A. j  e.  ( Idl `  R ) ( M 
 C_  j  ->  (
 j  =  M  \/  j  =  X )
 ) ) ) )
 
Theoremmaxidlidl 26019 A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  (
 ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  ->  M  e.  ( Idl `  R ) )
 
Theoremmaxidlnr 26020 A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  ->  M  =/=  X )
 
Theoremmaxidlmax 26021 A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  /\  ( I  e.  ( Idl `  R )  /\  M  C_  I ) ) 
 ->  ( I  =  M  \/  I  =  X ) )
 
Theoremmaxidln1 26022 One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)
 |-  H  =  ( 2nd `  R )   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  ->  -.  U  e.  M )
 
Theoremmaxidln0 26023 A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  ->  U  =/=  Z )
 
16.14.19  Prime rings and integral domains
 
Syntaxcprrng 26024 Extend class notation with the class of prime rings.
 class  PrRing
 
Syntaxcdmn 26025 Extend class notation with the class of domains.
 class  Dmn
 
Definitiondf-prrngo 26026 Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  PrRing  =  {
 r  e.  RingOps  |  {
 (GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r ) }
 
Definitiondf-dmn 26027 Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  Dmn  =  ( PrRing  i^i  Com2 )
 
Theoremisprrngo 26028 The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  PrRing  <->  ( R  e.  RingOps  /\  { Z }  e.  ( PrIdl `  R ) ) )
 
Theoremprrngorngo 26029 A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( R  e.  PrRing  ->  R  e. 
 RingOps )
 
Theoremsmprngopr 26030 A simple ring (one whose only ideals are  0 and  R) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  U  =/=  Z  /\  ( Idl `  R )  =  { { Z } ,  X } )  ->  R  e.  PrRing )
 
Theoremdivrngpr 26031 A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  ( R  e.  DivRingOps  ->  R  e.  PrRing )
 
Theoremisdmn 26032 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e.  Com2 )
 )
 
Theoremisdmn2 26033 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e. CRingOps ) )
 
Theoremdmncrng 26034 A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  ( R  e.  Dmn  ->  R  e. CRingOps )
 
Theoremdmnrngo 26035 A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  ( R  e.  Dmn  ->  R  e. 
 RingOps )
 
Theoremflddmn 26036 A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( K  e.  Fld  ->  K  e.  Dmn )
 
16.14.20  Ideal generators
 
Syntaxcigen 26037 Extend class notation with the ideal generation function.
 class  IdlGen
 
Definitiondf-igen 26038* Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  IdlGen  =  ( r  e.  RingOps ,  s  e.  ~P ran  ( 1st `  r )  |->  |^| { j  e.  ( Idl `  r
 )  |  s  C_  j } )
 
Theoremigenval 26039* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R )  |  S  C_  j } )
 
Theoremigenss 26040 A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  S  C_  X )  ->  S  C_  ( R  IdlGen  S ) )
 
Theoremigenidl 26041 The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  e.  ( Idl `  R ) )
 
Theoremigenmin 26042 The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  (
 ( R  e.  RingOps  /\  I  e.  ( Idl `  R )  /\  S  C_  I )  ->  ( R  IdlGen  S )  C_  I )
 
Theoremigenidl2 26043 The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  (
 ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  ->  ( R  IdlGen  I )  =  I )
 
Theoremigenval2 26044* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  S  C_  X )  ->  ( ( R  IdlGen  S )  =  I  <->  ( I  e.  ( Idl `  R )  /\  S  C_  I  /\  A. j  e.  ( Idl `  R ) ( S  C_  j  ->  I 
 C_  j ) ) ) )
 
Theoremprnc 26045* A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e. CRingOps  /\  A  e.  X )  ->  ( R  IdlGen  { A } )  =  { x  e.  X  |  E. y  e.  X  x  =  ( y H A ) } )
 
Theoremisfldidl 26046 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  K )   &    |-  H  =  ( 2nd `  K )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( K  e.  Fld  <->  ( K  e. CRingOps  /\  U  =/=  Z 
 /\  ( Idl `  K )  =  { { Z } ,  X }
 ) )
 
Theoremisfldidl2 26047 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  G  =  ( 1st `  K )   &    |-  H  =  ( 2nd `  K )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( K  e.  Fld  <->  ( K  e. CRingOps  /\  X  =/=  { Z }  /\  ( Idl `  K )  =  { { Z } ,  X } ) )
 
Theoremispridlc 26048* The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e. CRingOps  ->  ( P  e.  ( PrIdl `  R )  <->  ( P  e.  ( Idl `  R )  /\  P  =/=  X  /\  A. a  e.  X  A. b  e.  X  (
 ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P )
 ) ) ) )
 
Theorempridlc 26049 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R ) ) 
 /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  e.  P )
 )  ->  ( A  e.  P  \/  B  e.  P ) )
 
Theorempridlc2 26050 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R ) ) 
 /\  ( A  e.  ( X  \  P ) 
 /\  B  e.  X  /\  ( A H B )  e.  P )
 )  ->  B  e.  P )
 
Theorempridlc3 26051 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R ) ) 
 /\  ( A  e.  ( X  \  P ) 
 /\  B  e.  ( X  \  P ) ) )  ->  ( A H B )  e.  ( X  \  P ) )
 
Theoremisdmn3 26052* The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( R  e.  Dmn  <->  ( R  e. CRingOps  /\  U  =/=  Z 
 /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z )
 ) ) )
 
Theoremdmnnzd 26053 A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  =  Z ) ) 
 ->  ( A  =  Z  \/  B  =  Z ) )
 
Theoremdmncan1 26054 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  /\  A  =/=  Z )  ->  ( ( A H B )  =  ( A H C )  ->  B  =  C )
 )
 
Theoremdmncan2 26055 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  /\  C  =/=  Z )  ->  ( ( A H C )  =  ( B H C )  ->  A  =  B )
 )
 
16.15  Mathbox for Rodolfo Medina
 
16.15.1  Partitions
 
Theoremprtlem60 26056 Lemma for prter3 26103. (Contributed by Rodolfo Medina, 9-Oct-2010.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ps  ->  ( th  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
 
TheoremimpbiddOLD 26057 Lemma for prter3 26103. (Moved to impbidd 183 in main set.mm and may be deleted by mathbox owner, RM. --NM 15-May-2013.) (Contributed by Rodolfo Medina, 12-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )
 
Theorembicomdd 26058 Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( th  <->  ch ) ) )
 
TheorembicomddOLD 26059 Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( th  <->  ch ) ) )
 
Theoremprtlem1 26060 Add a disjunct in the antecedent. (Contributed by Rodolfo Medina, 24-Sep-2010.)
 |-  ( ps  ->  ( ch  ->  ph ) )   =>    |-  ( ( ph  \/  ps )  ->  ( ch  -> 
 ph ) )
 
Theoremjca2 26061 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ps  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremjca2r 26062 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ps  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( th  /\  ch ) ) )
 
Theoremjca3 26063 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  ( ch  /\  ta )
 ) ) )
 
Theoremprtlem50 26064 Lemma for prter3 26103. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ta )   =>    |-  ( ph  ->  (
 ( ps  /\  th )  ->  ( ch  /\  ta ) ) )
 
Theoreman43 26065 Rearrangement of 4 conjuncts. (Contributed by Rodolfo Medina, 24-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  (
 ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  ( ( ph  /\ 
 th )  /\  ( ps  /\  ch ) ) )
 
Theoreman43OLD 26066 Rearrangement of 4 conjuncts. (Contributed by Rodolfo Medina, 24-Sep-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  ( ( ph  /\ 
 th )  /\  ( ps  /\  ch ) ) )
 
Theoreman3 26067 A rearrangement of conjuncts. (Contributed by Rodolfo Medina, 25-Sep-2010.)
 |-  (
 ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  ->  ( ph  /\  th ) )
 
Theoremprtlem70 26068 Lemma for prter3 26103: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)
 |-  (
 ( ( ( ps 
 /\  et )  /\  (
 ( ph  /\  th )  /\  ( ch  /\  ta ) ) )  /\  ph )  <->  ( ( ph  /\  ( ps  /\  ( ch  /\  ( th  /\  ta ) ) ) ) 
 /\  et ) )
 
Theoremibdr 26069 Reverse of ibd. (Contributed by Rodolfo Medina, 30-Sep-2010.)
 |-  ( ph  ->  ( ch  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( ch  ->  ps ) )
 
Theorempm5.31r 26070 Variant of pm5.31 574. (Contributed by Rodolfo Medina, 15-Oct-2010.)
 |-  (
 ( ch  /\  ( ph  ->  ps ) )  ->  ( ph  ->  ( ch  /\ 
 ps ) ) )
 
Theoremexan3 26071 Cancel a conjunct from the scope of an existential quantifier. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( E. x ( ph  /\  ps )  ->  E. x ps )
 
Theoremexan3OLD 26072 Cancel a conjunct from the scope of an existential quantifier. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x ( ph  /\  ps )  ->  E. x ps )
 
Theorem2r19.29 26073 Double the quantifiers of theorem r19.29. (Contributed by Rodolfo Medina, 25-Sep-2010.)
 |-  (
 ( A. x  e.  A  A. y  e.  B  ph  /\ 
 E. x  e.  A  E. y  e.  B  ps )  ->  E. x  e.  A  E. y  e.  B  ( ph  /\  ps ) )
 
Theoremprtlem100 26074 Lemma for prter3 26103. (Contributed by Rodolfo Medina, 19-Oct-2010.)
 |-  ( E. x  e.  A  ( B  e.  x  /\  ph )  <->  E. x  e.  ( A  \  { (/) } )
 ( B  e.  x  /\  ph ) )
 
Theoremprtlem5 26075* Lemma for prter1 26100, prter2 26102, prter3 26103 and prtex 26101. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  ( [ s  /  v ] [ r  /  u ] E. x  e.  A  ( u  e.  x  /\  v  e.  x ) 
 <-> 
 E. x  e.  A  ( r  e.  x  /\  s  e.  x ) )
 
Theoremprtlem90 26076 Lemma for prter2 26102. (Contributed by Rodolfo Medina, 17-Oct-2010.)
 |-  ( -.  A  e.  B  ->  ( C  e.  B  ->  C  =/=  A ) )
 
Theoremprtlem80 26077 Lemma for prter2 26102. (Contributed by Rodolfo Medina, 17-Oct-2010.)
 |-  ( A  e.  B  ->  -.  A  e.  ( C 
 \  { A }
 ) )
 
Theoremn0el 26078* Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)
 |-  ( -.  (/)  e.  A  <->  A. x  e.  A  E. u  u  e.  x )
 
Theoremceqsex3OLD 26079* Version of ceqsex 2790 with an antecedent instead of a hypothesis. (Use ceqsexg 2867 instead of this one. --NM 13-Aug-11) (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  _V 
 ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
 
Theoremceqsex3vOLD 26080* Version of ceqsexv 2791 with an antecedent instead of a hypothesis. (Use ceqsexgv 2868 instead of this one. --NM 13-Aug-11) (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  _V 
 ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
 
TheoremeqrelrdvOLD 26081* Deduce equality of relations from equivalence of membership. (Moved to eqrelrdv 4757 in main set.mm and may be deleted by mathbox owner, RM. --NM 20-Feb-2014.) (Contributed by Rodolfo Medina, 10-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  Rel  A   &    |-  Rel 
 B   &    |-  ( ph  ->  ( <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqrelrdv2OLD 26082* Another version of eqrelrdv 4757. (Moved to eqrelrdv2 4760 in main set.mm and may be deleted by mathbox owner, RM. --NM 20-Feb-2014.) (Contributed by Rodolfo Medina, 30-Sep-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B ) )   =>    |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
 
Theorembrabsb2 26083* Closed form of brabsbOLD 4232. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( z R w  <->  [ w  /  y ] [ z  /  x ] ph ) )
 
Theoremeqbrrdv2 26084* Other version of eqbrrdiv 4759. (Contributed by Rodolfo Medina, 30-Sep-2010.)
 |-  (
 ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( x A y  <->  x B y ) )   =>    |-  ( ( ( Rel 
 A  /\  Rel  B ) 
 /\  ph )  ->  A  =  B )
 
Theoremprtlem9 26085* Lemma for prter3 26103. (Contributed by Rodolfo Medina, 25-Sep-2010.)
 |-  ( A  e.  B  ->  E. x  e.  B  [ x ]  .~  =  [ A ]  .~  )
 
Theoremprtlem10 26086* Lemma for prter3 26103. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  (  .~  Er  A  ->  (
 z  e.  A  ->  ( z  .~  w  <->  E. v  e.  A  ( z  e.  [ v ]  .~  /\  w  e. 
 [ v ]  .~  ) ) ) )
 
Theoremprtlem11 26087 Lemma for prter2 26102. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( B  e.  D  ->  ( C  e.  A  ->  ( B  =  [ C ]  .~  ->  B  e.  ( A /.  .~  )
 ) ) )
 
Theoremprtlem12 26088* Lemma for prtex 26101 and prter3 26103. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  (  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }  ->  Rel  .~  )
 
Theoremprtlem13 26089* Lemma for prter1 26100, prter2 26102, prter3 26103 and prtex 26101. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( z  .~  w 
 <-> 
 E. v  e.  A  ( z  e.  v  /\  w  e.  v
 ) )
 
Theoremprtlem16 26090* Lemma for prtex 26101, prter2 26102 and prter3 26103. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  dom  .~  =  U. A
 
Theoremprtlem400 26091* Lemma for prter2 26102 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  -.  (/)  e.  ( U. A /.  .~  )
 
Syntaxwprt 26092 Extend the definition of a wff to include the partition predicate.
 wff  Prt 
 A
 
Definitiondf-prt 26093* Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  ( Prt  A  <->  A. x  e.  A  A. y  e.  A  ( x  =  y  \/  ( x  i^i  y
 )  =  (/) ) )
 
Theoremerprt 26094 The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  (  .~  Er  X  ->  Prt  ( A /.  .~  ) )
 
Theoremprtlem14 26095* Lemma for prter1 26100, prter2 26102 and prtex 26101. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  ( Prt  A  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( w  e.  x  /\  w  e.  y )  ->  x  =  y ) ) )
 
Theoremprtlem15 26096* Lemma for prter1 26100 and prtex 26101. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  ( Prt  A  ->  ( E. x  e.  A  E. y  e.  A  ( ( u  e.  x  /\  w  e.  x )  /\  ( w  e.  y  /\  v  e.  y )
 )  ->  E. z  e.  A  ( u  e.  z  /\  v  e.  z ) ) )
 
Theoremprtlem17 26097* Lemma for prter2 26102. (Contributed by Rodolfo Medina, 15-Oct-2010.)
 |-  ( Prt  A  ->  ( ( x  e.  A  /\  z  e.  x )  ->  ( E. y  e.  A  ( z  e.  y  /\  w  e.  y )  ->  w  e.  x ) ) )
 
Theoremprtlem18 26098* Lemma for prter2 26102. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( Prt  A  ->  ( ( v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  <->  z  .~  w ) ) )
 
Theoremprtlem19 26099* Lemma for prter2 26102. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( Prt  A  ->  ( ( v  e.  A  /\  z  e.  v )  ->  v  =  [ z ]  .~  ) )
 
Theoremprter1 26100* Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( Prt  A  ->  .~  Er  U. A )
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