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Theorem List for Metamath Proof Explorer - 26701-26800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremigenidl2 26701 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 26702* 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 26703* 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 26704 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 26705 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 26706* 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 26707 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 26708 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 26709 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 26710* 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 26711 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 26712 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 26713 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 )
 )
 
18.16  Mathbox for Rodolfo Medina
 
18.16.1  Partitions
 
Theoremprtlem60 26714 Lemma for prter3 26761. (Contributed by Rodolfo Medina, 9-Oct-2010.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ps  ->  ( th  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
 
TheoremimpbiddOLD 26715 Lemma for prter3 26761. (Moved to impbidd 181 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 26716 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 26717 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 26718 Add a disjunct in the antecedent. (Contributed by Rodolfo Medina, 24-Sep-2010.)
 |-  ( ps  ->  ( ch  ->  ph ) )   =>    |-  ( ( ph  \/  ps )  ->  ( ch  -> 
 ph ) )
 
Theoremjca2 26719 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ps  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremjca2r 26720 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ps  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( th  /\  ch ) ) )
 
Theoremjca3 26721 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 26722 Lemma for prter3 26761. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ta )   =>    |-  ( ph  ->  (
 ( ps  /\  th )  ->  ( ch  /\  ta ) ) )
 
Theoreman43 26723 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 26724 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 26725 A rearrangement of conjuncts. (Contributed by Rodolfo Medina, 25-Sep-2010.)
 |-  (
 ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  ->  ( ph  /\  th ) )
 
Theoremprtlem70 26726 Lemma for prter3 26761: 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 26727 Reverse of ibd. (Contributed by Rodolfo Medina, 30-Sep-2010.)
 |-  ( ph  ->  ( ch  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( ch  ->  ps ) )
 
Theorempm5.31r 26728 Variant of pm5.31 571. (Contributed by Rodolfo Medina, 15-Oct-2010.)
 |-  (
 ( ch  /\  ( ph  ->  ps ) )  ->  ( ph  ->  ( ch  /\ 
 ps ) ) )
 
Theoremexan3 26729 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 26730 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 26731 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 26732 Lemma for prter3 26761. (Contributed by Rodolfo Medina, 19-Oct-2010.)
 |-  ( E. x  e.  A  ( B  e.  x  /\  ph )  <->  E. x  e.  ( A  \  { (/) } )
 ( B  e.  x  /\  ph ) )
 
Theoremprtlem5 26733* Lemma for prter1 26758, prter2 26760, prter3 26761 and prtex 26759. (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 26734 Lemma for prter2 26760. (Contributed by Rodolfo Medina, 17-Oct-2010.)
 |-  ( -.  A  e.  B  ->  ( C  e.  B  ->  C  =/=  A ) )
 
Theoremprtlem80 26735 Lemma for prter2 26760. (Contributed by Rodolfo Medina, 17-Oct-2010.)
 |-  ( A  e.  B  ->  -.  A  e.  ( C 
 \  { A }
 ) )
 
Theoremn0el 26736* 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 26737* Version of ceqsex 2824 with an antecedent instead of a hypothesis. (Use ceqsexg 2901 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 26738* Version of ceqsexv 2825 with an antecedent instead of a hypothesis. (Use ceqsexgv 2902 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 26739* Deduce equality of relations from equivalence of membership. (Moved to eqrelrdv 4785 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 26740* Another version of eqrelrdv 4785. (Moved to eqrelrdv2 4788 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 26741* Closed form of brabsbOLD 4276. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( z R w  <->  [ w  /  y ] [ z  /  x ] ph ) )
 
Theoremeqbrrdv2 26742* Other version of eqbrrdiv 4787. (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 26743* Lemma for prter3 26761. (Contributed by Rodolfo Medina, 25-Sep-2010.)
 |-  ( A  e.  B  ->  E. x  e.  B  [ x ]  .~  =  [ A ]  .~  )
 
Theoremprtlem10 26744* Lemma for prter3 26761. (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 26745 Lemma for prter2 26760. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( B  e.  D  ->  ( C  e.  A  ->  ( B  =  [ C ]  .~  ->  B  e.  ( A /.  .~  )
 ) ) )
 
Theoremprtlem12 26746* Lemma for prtex 26759 and prter3 26761. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  (  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }  ->  Rel  .~  )
 
Theoremprtlem13 26747* Lemma for prter1 26758, prter2 26760, prter3 26761 and prtex 26759. (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 26748* Lemma for prtex 26759, prter2 26760 and prter3 26761. (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 26749* Lemma for prter2 26760 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 26750 Extend the definition of a wff to include the partition predicate.
 wff  Prt 
 A
 
Definitiondf-prt 26751* 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 26752 The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  (  .~  Er  X  ->  Prt  ( A /.  .~  ) )
 
Theoremprtlem14 26753* Lemma for prter1 26758, prter2 26760 and prtex 26759. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  ( Prt  A  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( w  e.  x  /\  w  e.  y )  ->  x  =  y ) ) )
 
Theoremprtlem15 26754* Lemma for prter1 26758 and prtex 26759. (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 26755* Lemma for prter2 26760. (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 26756* Lemma for prter2 26760. (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 26757* Lemma for prter2 26760. (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 26758* 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 )
 
Theoremprtex 26759* The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (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  ->  (  .~  e.  _V  <->  A  e.  _V ) )
 
Theoremprter2 26760* The quotient set of the equivalence relation generated by a partition equals the partition itself. (Contributed by Rodolfo Medina, 17-Oct-2010.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( Prt  A  ->  ( U. A /.  .~  )  =  ( A 
 \  { (/) } )
 )
 
Theoremprter3 26761* For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( ( S  Er  U. A  /\  ( U. A /. S )  =  ( A  \  { (/) } ) ) 
 ->  .~  =  S )
 
18.17  Mathbox for Stefan O'Rear
 
18.17.1  Additional elementary logic and set theory
 
Theoremnelss 26762 Demonstrate by witnesses that two classes lack a subclass relation. (Contributed by Stefan O'Rear, 5-Feb-2015.)
 |-  (
 ( A  e.  B  /\  -.  A  e.  C )  ->  -.  B  C_  C )
 
Theoremmoxfr 26763* Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)
 |-  A  e.  _V   &    |-  E! y  x  =  A   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( E* x ph  <->  E* y ps )
 
Theoremraldifsni 26764 Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |-  ( A. x  e.  ( A  \  { B }
 )  -.  ph  <->  A. x  e.  A  ( ph  ->  x  =  B ) )
 
18.17.2  Additional theory of functions
 
Theoremfninfp 26765* Express the class of fixed points of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( F  Fn  A  ->  dom  ( F  i^i  _I  )  =  { x  e.  A  |  ( F `  x )  =  x }
 )
 
Theoremfnelfp 26766 Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  (
 ( F  Fn  A  /\  X  e.  A ) 
 ->  ( X  e.  dom  ( F  i^i  _I  )  <->  ( F `  X )  =  X ) )
 
Theoremfndifnfp 26767* Express the class of non-fixed points of a function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
 
Theoremfnelnfp 26768 Property of a non-fixed point of a function. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  (
 ( F  Fn  A  /\  X  e.  A ) 
 ->  ( X  e.  dom  ( F  \  _I  )  <->  ( F `  X )  =/=  X ) )
 
Theoremfnnfpeq0 26769 A function is the identity iff it moves no points. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  ( F  Fn  A  ->  ( dom  ( F  \  _I  )  =  (/)  <->  F  =  (  _I  |`  A ) ) )
 
Theoremimaiinfv 26770* Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (
 ( F  Fn  A  /\  B  C_  A )  -> 
 |^|_ x  e.  B  ( F `  x )  =  |^| ( F " B ) )
 
Theoremralxpxfr2d 26771* Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |-  A  e.  _V   &    |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  E. z  e.  D  x  =  A )
 )   &    |-  ( ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  B  ps 
 <-> 
 A. y  e.  C  A. z  e.  D  ch ) )
 
Theoremralxpmap 26772* Quantification over functions in terms of quantification over values and punctured functions. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  (
 f  =  ( g  u.  { <. J ,  y >. } )  ->  ( ph  <->  ps ) )   =>    |-  ( J  e.  T  ->  ( A. f  e.  ( S  ^m  T ) ph  <->  A. y  e.  S  A. g  e.  ( S 
 ^m  ( T  \  { J } ) ) ps ) )
 
Theoremfunsnfsup 26773 Finite support for a function extended by a singleton. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |-  (
 ( `' ( F  u.  { <. X ,  Y >. } ) " Z )  e.  Fin  <->  ( `' F " Z )  e.  Fin )
 
Theoremfvtresfn 26774* Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )   =>    |-  ( X  e.  B  ->  ( F `  X )  =  ( X  |`  V ) )
 
18.17.3  Extensions beyond function theory
 
Theoremgsumvsmul 26775* Pull a scalar multiplication out of a sum of vectors. EDITORIAL: properly generalizes gsummulc2 15393, since every ring is a left module over itself. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  B  =  ( Base `  R )   &    |-  S  =  (Scalar `  R )   &    |-  K  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .+  =  ( +g  `  R )   &    |-  .x.  =  ( .s `  R )   &    |-  ( ph  ->  R  e.  LMod )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  X  e.  K )   &    |-  (
 ( ph  /\  k  e.  A )  ->  Y  e.  B )   &    |-  ( ph  ->  ( `' ( k  e.  A  |->  Y ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   =>    |-  ( ph  ->  ( R  gsumg  ( k  e.  A  |->  ( X  .x.  Y ) ) )  =  ( X  .x.  ( R  gsumg  (
 k  e.  A  |->  Y ) ) ) )
 
TheoremgrpinvnzOLD 26776 The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.) . (Moved to grpinvnz 14541 in main set.mm and may be deleted by mathbox owner, SO. --NM 23-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  (
 ( G  e.  Grp  /\  X  e.  B  /\  X  =/=  .0.  )  ->  ( N `  X )  =/=  .0.  )
 
TheoremgrpinvnzclOLD 26777 The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.) . (Moved to grpinvnzcl 14542 in main set.mm and may be deleted by mathbox owner, SO. --NM 23-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  (
 ( G  e.  Grp  /\  X  e.  ( B 
 \  {  .0.  }
 ) )  ->  ( N `  X )  e.  ( B  \  {  .0.  } ) )
 
Theoremlcomf 26778 A linear-combination sum is a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  B  =  ( Base `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G : I --> K )   &    |-  ( ph  ->  H : I
 --> B )   &    |-  ( ph  ->  I  e.  V )   =>    |-  ( ph  ->  ( G  o F  .x.  H ) : I --> B )
 
Theoremlcomfsup 26779 A linear-combination sum is finitely supported if the coefficients are. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  B  =  ( Base `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G : I --> K )   &    |-  ( ph  ->  H : I
 --> B )   &    |-  ( ph  ->  I  e.  V )   &    |-  .0.  =  ( 0g `  W )   &    |-  Y  =  ( 0g
 `  F )   &    |-  ( ph  ->  ( `' G " ( _V  \  { Y } ) )  e. 
 Fin )   =>    |-  ( ph  ->  ( `' ( G  o F  .x.  H ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )
 
18.17.4  Additional topology
 
Theoremelrfi 26780* Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (
 ( B  e.  V  /\  C  C_  ~P B )  ->  ( A  e.  ( fi `  ( { B }  u.  C ) )  <->  E. v  e.  ( ~P C  i^i  Fin ) A  =  ( B  i^i  |^| v ) ) )
 
Theoremelrfirn 26781* Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (
 ( B  e.  V  /\  F : I --> ~P B )  ->  ( A  e.  ( fi `  ( { B }  u.  ran  F ) )  <->  E. v  e.  ( ~P I  i^i  Fin ) A  =  ( B  i^i  |^|_ y  e.  v  ( F `  y ) ) ) )
 
Theoremelrfirn2 26782* Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (
 ( B  e.  V  /\  A. y  e.  I  C  C_  B )  ->  ( A  e.  ( fi `  ( { B }  u.  ran  ( y  e.  I  |->  C ) ) )  <->  E. v  e.  ( ~P I  i^i  Fin ) A  =  ( B  i^i  |^|_ y  e.  v  C ) ) )
 
Theoremcmpfiiin 26783* In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  X  =  U. J   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ( ph  /\  k  e.  I ) 
 ->  S  e.  ( Clsd `  J ) )   &    |-  (
 ( ph  /\  ( l 
 C_  I  /\  l  e.  Fin ) )  ->  ( X  i^i  |^|_ k  e.  l  S )  =/=  (/) )   =>    |-  ( ph  ->  ( X  i^i  |^|_ k  e.  I  S )  =/=  (/) )
 
18.17.5  Characterization of closure operators. Kuratowski closure axioms
 
Theoremismrcd1 26784* Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 13521), isotone (satisfies mrcss 13520), and idempotent (satisfies mrcidm 13523) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 26785 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  F : ~P B --> ~P B )   &    |-  ( ( ph  /\  x  C_  B )  ->  x  C_  ( F `  x ) )   &    |-  ( ( ph  /\  x  C_  B  /\  y  C_  x )  ->  ( F `  y ) 
 C_  ( F `  x ) )   &    |-  (
 ( ph  /\  x  C_  B )  ->  ( F `
  ( F `  x ) )  =  ( F `  x ) )   =>    |-  ( ph  ->  dom  ( F  i^i  _I  )  e.  (Moore `  B )
 )
 
Theoremismrcd2 26785* Second half of ismrcd1 26784. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  F : ~P B --> ~P B )   &    |-  ( ( ph  /\  x  C_  B )  ->  x  C_  ( F `  x ) )   &    |-  ( ( ph  /\  x  C_  B  /\  y  C_  x )  ->  ( F `  y ) 
 C_  ( F `  x ) )   &    |-  (
 ( ph  /\  x  C_  B )  ->  ( F `
  ( F `  x ) )  =  ( F `  x ) )   =>    |-  ( ph  ->  F  =  (mrCls `  dom  ( F  i^i  _I  ) ) )
 
Theoremistopclsd 26786* A closure function which satisfies sscls 16795, clsidm 16806, cls0 16819, and clsun 26257 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  F : ~P B --> ~P B )   &    |-  ( ( ph  /\  x  C_  B )  ->  x  C_  ( F `  x ) )   &    |-  ( ( ph  /\  x  C_  B )  ->  ( F `  ( F `  x ) )  =  ( F `  x ) )   &    |-  ( ph  ->  ( F `  (/) )  =  (/) )   &    |-  (
 ( ph  /\  x  C_  B  /\  y  C_  B )  ->  ( F `  ( x  u.  y
 ) )  =  ( ( F `  x )  u.  ( F `  y ) ) )   &    |-  J  =  { z  e.  ~P B  |  ( F `  ( B 
 \  z ) )  =  ( B  \  z ) }   =>    |-  ( ph  ->  ( J  e.  (TopOn `  B )  /\  ( cls `  J )  =  F ) )
 
Theoremismrc 26787* A function is a Moore closure operator iff it satisfies mrcssid 13521, mrcss 13520, and mrcidm 13523. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( F  e.  (mrCls " (Moore `  B ) )  <->  ( B  e.  _V 
 /\  F : ~P B
 --> ~P B  /\  A. x A. y ( ( x  C_  B  /\  y  C_  x )  ->  ( x  C_  ( F `
  x )  /\  ( F `  y ) 
 C_  ( F `  x )  /\  ( F `
  ( F `  x ) )  =  ( F `  x ) ) ) ) )
 
18.17.6  Algebraic closure systems
 
Syntaxcnacs 26788 Class of Noetherian closure systems.
 class NoeACS
 
Definitiondf-nacs 26789* Define a closure system of Noetherian type (not standard terminology) as an algebraic system where all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |- NoeACS  =  ( x  e.  _V  |->  { c  e.  (ACS `  x )  |  A. s  e.  c  E. g  e.  ( ~P x  i^i  Fin ) s  =  ( (mrCls `  c ) `  g ) } )
 
Theoremisnacs 26790* Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (NoeACS `  X ) 
 <->  ( C  e.  (ACS `  X )  /\  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g ) ) )
 
Theoremnacsfg 26791* In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  (
 ( C  e.  (NoeACS `  X )  /\  S  e.  C )  ->  E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g ) )
 
Theoremisnacs2 26792 Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (NoeACS `  X ) 
 <->  ( C  e.  (ACS `  X )  /\  ( F " ( ~P X  i^i  Fin ) )  =  C ) )
 
Theoremmrefg2 26793* Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  ( E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g )  <->  E. g  e.  ( ~P S  i^i  Fin ) S  =  ( F `  g ) ) )
 
Theoremmrefg3 26794* Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  (
 ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  ( E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g )  <->  E. g  e.  ( ~P S  i^i  Fin ) S  C_  ( F `  g ) ) )
 
Theoremnacsacs 26795 A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  ( C  e.  (NoeACS `  X )  ->  C  e.  (ACS `  X ) )
 
Theoremisnacs3 26796* A choice-free order equivalent to the Noetherian condition on a closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  ( C  e.  (NoeACS `  X ) 
 <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C ( (toInc `  s )  e. Dirset  ->  U. s  e.  s
 ) ) )
 
Theoremincssnn0 26797* Transitivity induction of subsets, lemma for nacsfix 26798. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  (
 ( A. x  e.  NN0  ( F `  x ) 
 C_  ( F `  ( x  +  1
 ) )  /\  A  e.  NN0  /\  B  e.  ( ZZ>= `  A )
 )  ->  ( F `  A )  C_  ( F `  B ) )
 
Theoremnacsfix 26798* An increasing sequence of closed sets in a Noetherian-type closure system eventually fixates. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  (
 ( C  e.  (NoeACS `  X )  /\  F : NN0 --> C  /\  A. x  e.  NN0  ( F `
  x )  C_  ( F `  ( x  +  1 ) ) )  ->  E. y  e.  NN0  A. z  e.  ( ZZ>=
 `  y ) ( F `  z )  =  ( F `  y ) )
 
18.17.7  Miscellanea 1. Map utilities
 
Theoremconstmap 26799 A constant (represented without dummy variables) is an element of a function set.

_Note: In the following development, we will be quite often quantifying over functions and points in N-dimensional space (which are equivalent to functions from an "index set"). Many of the following theorems exist to transfer standard facts about functions to elements of function sets._ (Contributed by Stefan O'Rear, 30-Aug-2014.) (Revised by Stefan O'Rear, 5-May-2015.)

 |-  A  e.  _V   &    |-  C  e.  _V   =>    |-  ( B  e.  C  ->  ( A  X.  { B } )  e.  ( C  ^m  A ) )
 
Theoremelmapfun 26800 A mapping is always a function. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  ( A  e.  ( B  ^m  C )  ->  Fun  A )
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