HomeHome Metamath Proof Explorer
Theorem List (p. 238 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

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

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21490)
  Hilbert Space Explorer  Hilbert Space Explorer
(21491-23013)
  Users' Mathboxes  Users' Mathboxes
(23014-31421)
 

Theorem List for Metamath Proof Explorer - 23701-23800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelno 23701* Membership in the surreals. (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
 |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
 
Theoremsltval 23702* The value of the surreal less than relationship. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
 <-> 
 E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `
  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `
  x ) ) ) )
 
Theorembdayval 23703 The value of the birthday function within the surreals. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ( A  e.  No  ->  (
 bday `  A )  = 
 dom  A )
 
Theoremnofun 23704 A surreal is a function. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  Fun 
 A )
 
Theoremnodmon 23705 The domain of a surreal is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  dom 
 A  e.  On )
 
Theoremnorn 23706 The range of a surreal is a subset of the surreal signs. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  ran 
 A  C_  { 1o ,  2o } )
 
Theoremnodmord 23707 The domain of a surreal has the ordinal property. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  Ord 
 dom  A )
 
Theoremelno2 23708 An alternative condition for membership in  No. (Contributed by Scott Fenton, 21-Mar-2012.)
 |-  ( A  e.  No  <->  ( Fun  A  /\  dom  A  e.  On  /\ 
 ran  A  C_  { 1o ,  2o } ) )
 
Theoremelno3 23709 Another condition for membership in 
No. (Contributed by Scott Fenton, 14-Apr-2012.)
 |-  ( A  e.  No  <->  ( A : dom  A --> { 1o ,  2o }  /\  dom  A  e.  On ) )
 
Theoremsltval2 23710* Alternate expression for surreal less than. Two surreals obey surreal less than iff they obey the sign ordering at the first place they differ. (Contributed by Scott Fenton, 17-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
 <->  ( A `  |^| { a  e.  On  |  ( A `
  a )  =/=  ( B `  a
 ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `
  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) } ) ) )
 
Theoremnofv 23711 The function value of a surreal is either a sign or the empty set. (Contributed by Scott Fenton, 22-Jun-2011.)
 |-  ( A  e.  No  ->  ( ( A `  X )  =  (/)  \/  ( A `  X )  =  1o  \/  ( A `
  X )  =  2o ) )
 
Theoremnosgnn0 23712  (/) is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  -.  (/) 
 e.  { 1o ,  2o }
 
Theoremnosgnn0i 23713 If  X is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   =>    |-  (/) 
 =/=  X
 
Theoremnoreson 23714 The restriction of a surreal to an ordinal is still a surreal. (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  On )  ->  ( A  |`  B )  e.  No )
 
Theoremsltsgn1 23715* If  A < s B, then the sign of  A at the first place they differ is either undefined or  1o (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  ( ( A `
  |^| { k  e. 
 On  |  ( A `
  k )  =/=  ( B `  k
 ) } )  =  (/)  \/  ( A `  |^|
 { k  e.  On  |  ( A `  k
 )  =/=  ( B `  k ) } )  =  1o ) ) )
 
Theoremsltsgn2 23716* If  A < s B, then the sign of  B at the first place they differ is either undefined or  2o (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  ( ( B `
  |^| { k  e. 
 On  |  ( A `
  k )  =/=  ( B `  k
 ) } )  =  (/)  \/  ( B `  |^|
 { k  e.  On  |  ( A `  k
 )  =/=  ( B `  k ) } )  =  2o ) ) )
 
Theoremsltintdifex 23717* If  A < s B, then the intersection of all the ordinals that have differing signs in  A and  B exists. (Contributed by Scott Fenton, 22-Feb-2012.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) }  e.  _V ) )
 
Theoremsltres 23718 Lemma for axfe (future) . If the restrictions of two surreals to a given ordinal obey surreal less than, then so do the two surreals themselves. (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  ( ( A  |`  X ) < s ( B  |`  X )  ->  A < s B ) )
 
Theoremnoxpsgn 23719 The cross product of an ordinal and the singleton of a sign is a surreal. (Contributed by Scott Fenton, 21-Jun-2011.)
 |-  X  e.  { 1o ,  2o }   =>    |-  ( A  e.  On  ->  ( A  X.  { X } )  e.  No )
 
Theoremnoxp1o 23720 The cross product of an ordinal and  { 1o } is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.)
 |-  ( A  e.  On  ->  ( A  X.  { 1o } )  e.  No )
 
Theoremnoxp2o 23721 The cross product of an ordinal and  { 2o } is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.)
 |-  ( A  e.  On  ->  ( A  X.  { 2o } )  e.  No )
 
18.7.23  Surreal Numbers: Ordering
 
Theoremaxsltsolem1 23722 Lemma for axsltso 23723. The sign expansion relationship totally orders the surreal signs. (Contributed by axsltsolem1, 8-Jun-2011.)
 |-  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  ( { 1o ,  2o }  u.  { (/)
 } )
 
Theoremaxsltso 23723 Surreal less than totally orders the surreals. Alling's axiom (O). (Contributed by Scott Fenton, 9-Jun-2011.)
 |-  < s  Or  No
 
Theoremsltirr 23724 Surreal less than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  -.  A < s A )
 
Theoremslttr 23725 Surreal less than is transitive. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No  /\  C  e.  No )  ->  ( ( A <
 s B  /\  B < s C )  ->  A < s C ) )
 
Theoremsltasym 23726 Surreal less than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  -.  B < s A ) )
 
Theoremslttri 23727 Surreal less than obeys trichotomy. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  \/  A  =  B  \/  B < s A ) )
 
Theoremslttrieq2 23728 Trichotomy law for surreal less than. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A  =  B  <->  ( -.  A < s B  /\  -.  B <
 s A ) ) )
 
18.7.24  Surreal Numbers: Birthday Function
 
Theoremaxbday 23729 The birthday function maps the surreals onto the ordinals. Alling's axiom (B). (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
 |-  bday : No -onto-> On
 
Theorembdayfun 23730 The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  Fun  bday
 
Theorembdayrn 23731 The birthday function's range is 
On (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ran  bday 
 =  On
 
Theorembdaydm 23732 The birthday function's domain is 
No (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  dom  bday 
 =  No
 
Theorembdayfn 23733 The birthday function is a function over  No (Contributed by Scott Fenton, 30-Jun-2011.)
 |-  bday  Fn 
 No
 
Theorembdayelon 23734 The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ( bday `  A )  e. 
 On
 
Theoremnoprc 23735 The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  -.  No  e.  _V
 
18.7.25  Surreal Numbers: Density
 
Theoremaxdenselem1 23736 Lemma for axdense 23744. A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  A  Fn  ( bday `  A ) )
 
Theoremaxdenselem2 23737 Lemma for axfe (future) and axdense 23744. The value of a surreal at its birthday is  (/). (Shortened proof on 2012-Apr-14, SF) (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
 
Theoremaxdenselem3 23738* Lemma for axdense 23744. If one surreal is older than another, then there is an ordinal at which they are not equal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A )  e.  ( bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
 
Theoremaxdenselem4 23739* Lemma for axdense 23744. Show that a particular abstraction is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  |^| { a  e.  On  |  ( A `
  a )  =/=  ( B `  a
 ) }  e.  On )
 
Theoremaxdenselem5 23740* Lemma for axdense 23744. If the birthdays of two distinct surreals are equal, then the ordinal from axdenselem4 23739 is an element of that birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
 bday `  B )  /\  A < s B ) )  ->  |^| { a  e.  On  |  ( A `
  a )  =/=  ( B `  a
 ) }  e.  ( bday `  A ) )
 
Theoremaxdenselem6 23741* The restriction of a surreal to the abstraction from axdenselem4 23739 is still a surreal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
 bday `  B )  /\  A < s B ) )  ->  ( A  |` 
 |^| { a  e.  On  |  ( A `  a
 )  =/=  ( B `  a ) } )  e.  No )
 
Theoremaxdenselem7 23742* Lemma for axdense 23744. 
A and  B are equal at all elements of the abstraction. (Contributed by Scott Fenton, 17-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
 bday `  B )  /\  A < s B ) )  ->  ( C  e.  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) }  ->  ( A `  C )  =  ( B `  C ) ) )
 
Theoremaxdenselem8 23743* Lemma for axdense 23744. Give a condition for surreal less than when two surreals have the same birthday. (Contributed by Scott Fenton, 19-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B )
 )  ->  ( A < s B  <->  ( ( A `
  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) } )  =  1o  /\  ( B `
  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) } )  =  2o ) ) )
 
Theoremaxdense 23744* Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Alling's axiom (SD) (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
 bday `  B )  /\  A < s B ) )  ->  E. x  e.  No  ( ( bday `  x )  e.  ( bday `  A )  /\  A < s x  /\  x < s B ) )
 
Theoremnocvxminlem 23745* Lemma for nocvxmin 23746. Given two birthday-minimal elements of a convex class of surreals, they are not comparable. (Contributed by Scott Fenton, 30-Jun-2011.)
 |-  (
 ( A  C_  No  /\ 
 A. x  e.  A  A. y  e.  A  A. z  e.  No  (
 ( x < s
 z  /\  z < s y )  ->  z  e.  A ) )  ->  ( ( ( X  e.  A  /\  Y  e.  A )  /\  (
 ( bday `  X )  =  |^| ( bday " A )  /\  ( bday `  Y )  =  |^| ( bday " A ) ) ) 
 ->  -.  X < s Y ) )
 
Theoremnocvxmin 23746* Given a non-empty convex class of surreals, there is a unique birthday-minimal element of that class. (Contributed by Scott Fenton, 30-Jun-2011.)
 |-  (
 ( A  =/=  (/)  /\  A  C_ 
 No  /\  A. x  e.  A  A. y  e.  A  A. z  e. 
 No  ( ( x < s z  /\  z < s y ) 
 ->  z  e.  A ) )  ->  E! w  e.  A  ( bday `  w )  =  |^| ( bday " A ) )
 
18.7.26  Surreal Numbers: Full-Eta Property
 
Theoremaxfelem1 23747 Lemma for axfe (future) . The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.)
 |-  ( A  e.  V  ->  suc  U. ( bday " A )  e. 
 On )
 
Theoremaxfelem2 23748* Lemma for axfe (future) . Show a particular abstraction is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }   =>    |-  (
 ( F  C_  No  /\  F  e.  A ) 
 ->  C  e.  On )
 
Theoremaxfelem3 23749* Lemma for axfe (future) . Calculate the birthday of  ( C  X.  { X } ). (Contributed by Scott Fenton, 17-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }   =>    |-  (
 ( F  C_  No  /\  F  e.  A ) 
 ->  ( bday `  ( C  X.  { X } )
 )  =  C )
 
Theoremaxfelem4 23750* Lemma for axfe (future) . The infimum of the class of all ordinals such that  A is not  X is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   =>    |-  ( A  e.  No  -> 
 |^| { x  e.  On  |  ( A `  x )  =/=  X }  e.  On )
 
Theoremaxfelem5 23751* Lemma for axfe (future) . There is always a minimal value of a surreal that is not a given sign. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   =>    |-  ( A  e.  No  ->  ( A `  |^| { x  e.  On  |  ( A `
  x )  =/= 
 X } )  =/= 
 X )
 
Theoremaxfelem6 23752* Lemma for axfe (future) . Given an element  A of  F, then the first position where it differs from  X is strictly less than  C (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }   =>    |-  (
 ( F  C_  No  /\  A  e.  F ) 
 ->  |^| { x  e. 
 On  |  ( A `
  x )  =/= 
 X }  e.  C )
 
Theoremaxfelem7 23753* Lemma for axfe (future) . Calculate the value of  ( C  X.  { X } ) at the minimal ordinal whose value is different from  X. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }   =>    |-  (
 ( F  C_  No  /\  A  e.  F ) 
 ->  ( ( C  X.  { X } ) `  |^|
 { x  e.  On  |  ( A `  x )  =/=  X } )  =  X )
 
Theoremaxfelem8 23754* Lemma for axfe (future) . Bound the birthday of  ( C  X.  { X } ) above. (Contributed by Scott Fenton, 17-Aug-2011.)
 |-  S  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  S }   =>    |-  (
 ( ( F  C_  No  /\  F  e.  A )  /\  ( X  e.  On  /\  A. n  e.  F  ( bday `  n )  e.  X )
 )  ->  ( bday `  ( C  X.  { S } ) )  C_  X )
 
Theoremaxfelem9 23755* Lemma for axfe (future) . The full statement of the axiom when  R is empty. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  ( R  =  (/)  ->  (
 ( ( ( L 
 C_  No  /\  R  C_  No )  /\  ( L  e.  A  /\  R  e.  B ) )  /\  ( X  e.  On  /\ 
 A. n  e.  ( L  u.  R ) (
 bday `  n )  e.  X )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  E. z  e.  No  ( ( bday `  z )  C_  X  /\  A. x  e.  L  x < s z  /\  A. y  e.  R  z < s y ) ) )
 
Theoremaxfelem10 23756* Lemma for axfe (future) . The full statement of the axiom when  L is empty. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  ( L  =  (/)  ->  (
 ( ( ( L 
 C_  No  /\  R  C_  No )  /\  ( L  e.  A  /\  R  e.  B ) )  /\  ( X  e.  On  /\ 
 A. n  e.  ( L  u.  R ) (
 bday `  n )  e.  X )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  E. z  e.  No  ( ( bday `  z )  C_  X  /\  A. x  e.  L  x < s z  /\  A. y  e.  R  z < s y ) ) )
 
Theoremaxfelem11 23757* Lemma for axfe (future) . We are now going to construct a new surreal for the case where  L and  R are not null. The birthday/domain of this surreal will be the intersection of  D. This lemma just changes bound variables for later use. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   =>    |-  D  =  { b  e.  On  |  A. n  e.  L  A. m  e.  R  ( n  |`  b )  =/=  ( m  |`  b ) }
 
Theoremaxfelem12 23758* Lemma for axfe (future) . The surreal mentioned in the previous lemma will ultimately be the union of  C. Again, we just change bound variables for later. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  C  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  P  ( ( g  |`  a )  =  f 
 /\  ( h  |`  a )  =  f
 ) }   =>    |-  C  =  { b  |  E. c  e.  L  E. d  e.  R  E. e  e.  P  ( ( c  |`  e )  =  b  /\  ( d  |`  e )  =  b ) }
 
Theoremaxfelem13 23759 Lemma for axfe (future) . Establish an equality result for restriction of surreals. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  (
 ( A  C_  No  /\  X  e.  A ) 
 ->  ( X  |`  suc  U. ( bday " ( A  u.  B ) ) )  =  X )
 
Theoremaxfelem14 23760* Lemma for axfe (future) . Now, we prove that  P, the birthday of a surreal we are going to be working with, is actually an ordinal. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   =>    |-  ( ( ( ( L  C_  No  /\  R  C_ 
 No )  /\  ( L  e.  A  /\  R  e.  B )
 )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  P  e.  On )
 
Theoremaxfelem15 23761* Lemma for axfe (future) . For non-empty  L and  R,  P is non-empty. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   =>    |-  ( ( L  =/=  (/)  /\  R  =/=  (/) )  ->  P  =/=  (/) )
 
Theoremaxfelem16 23762* Lemma for axfe (future) . Preliminary work for axfelem17 23763. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  (
 ( ( ( A 
 C_  No  /\  B  C_  No )  /\  A. x  e.  A  A. y  e.  B  x < s
 y )  /\  ( X  e.  On  /\  (
 ( P  e.  A  /\  Q  e.  B ) 
 /\  ( R  e.  A  /\  S  e.  B ) ) )  /\  ( ( P  |`  X )  =  ( Q  |`  X ) 
 /\  ( R  |`  X )  =  ( S  |`  X ) ) )  ->  ( -.  ( P  |`  X ) < s ( R  |`  X )  /\  -.  ( Q  |`  X ) < s ( S  |`  X ) ) )
 
Theoremaxfelem17 23763* Lemma for axfe (future) . A uniqueness result for restrictions of surreals. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  (
 ( ( ( A 
 C_  No  /\  B  C_  No )  /\  A. x  e.  A  A. y  e.  B  x < s
 y )  /\  ( X  e.  On  /\  (
 ( P  e.  A  /\  Q  e.  B ) 
 /\  ( R  e.  A  /\  S  e.  B ) ) )  /\  ( ( P  |`  X )  =  ( Q  |`  X ) 
 /\  ( R  |`  X )  =  ( S  |`  X ) ) )  ->  (
 ( P  |`  X )  =  ( R  |`  X ) 
 /\  ( Q  |`  X )  =  ( S  |`  X ) ) )
 
Theoremaxfelem18 23764* Lemma for axfe (future) . At this point, we introduce a new surreal number  M (for "middle"). The next several lemmas prove that either  M or a closely related surreal has the required properties for the final theorem. We begin by calculating the domain of  M. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   &    |-  C  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  P  ( ( g  |`  a )  =  f  /\  ( h  |`  a )  =  f ) }   &    |-  M  =  U. C   =>    |-  ( ( ( ( L  C_  No  /\  R  C_ 
 No )  /\  ( L  e.  A  /\  R  e.  B )
 )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  dom  M  =  U. P )
 
Theoremaxfelem19 23765* Lemma for axfe (future) .  C is a subset of the surreals. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   &    |-  C  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  P  ( ( g  |`  a )  =  f  /\  ( h  |`  a )  =  f ) }   &    |-  M  =  U. C   =>    |-  ( ( ( ( L  C_  No  /\  R  C_ 
 No )  /\  ( L  e.  A  /\  R  e.  B )
 )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  C  C_ 
 No )
 
Theoremaxfelem20 23766* Lemma for axfe (future) . Next, we prove that  M is a function. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   &    |-  C  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  P  ( ( g  |`  a )  =  f  /\  ( h  |`  a )  =  f ) }   &    |-  M  =  U. C   =>    |-  ( ( ( ( L  C_  No  /\  R  C_ 
 No )  /\  ( L  e.  A  /\  R  e.  B )
 )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  Fun  M )
 
Theoremaxfelem21 23767* Lemma for axfe (future) . Now, using the previous three lemmas, we show that  M is a surreal. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   &    |-  C  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  P  ( ( g  |`  a )  =  f  /\  ( h  |`  a )  =  f ) }   &    |-  M  =  U. C   =>    |-  ( ( ( ( L  C_  No  /\  R  C_ 
 No )  /\  ( L  e.  A  /\  R  e.  B )
 )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  M  e.  No )
 
Theoremaxfelem22 23768* Lemma for axfe (future) . Given an element  K of  P, there are elements  l and  r of  L and  R whose restrictions to  K are a prefix of  M. (Contributed by Scott Fenton, 23-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   &    |-  C  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  P  ( ( g  |`  a )  =  f  /\  ( h  |`  a )  =  f ) }   &    |-  M  =  U. C   =>    |-  ( ( ( ( L  C_  No  /\  R  C_ 
 No )  /\  ( L  e.  A  /\  R  e.  B )
 )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  ( K  e.  P  ->  ( E. l  e.  L  ( l  |`  K ) 
 C_  M  /\  E. r  e.  R  (
 r  |`  K )  C_  M ) ) )
 
18.7.27  Symmetric difference
 
Syntaxcsymdif 23769 Declare the syntax for symmetric difference.
 class  ( A(++)
 B )
 
Definitiondf-symdif 23770 Define the symmetric difference of two classes. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  ( A(++) B )  =  ( ( A  \  B )  u.  ( B  \  A ) )
 
Theoremsymdifcom 23771 Symmetric difference commutes. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A(++) B )  =  ( B(++) A )
 
Theoremsymdifeq1 23772 Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A  =  B  ->  ( A(++) C )  =  ( B(++) C ) )
 
Theoremsymdifeq2 23773 Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A  =  B  ->  ( C(++) A )  =  ( C(++) B ) )
 
Theoremnfsymdif 23774 Hypothesis builder for symmetric difference. (Contributed by Scott Fenton, 19-Feb-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A(++) B )
 
Theoremelsymdif 23775 Membership in a symmetric difference. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  ( A  e.  ( B(++) C )  <->  -.  ( A  e.  B 
 <->  A  e.  C ) )
 
Theoremsymdif0 23776 Symmetric difference with the empty class. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A(++) (/) )  =  A
 
TheoremsymdifV 23777 Symmetric difference with the universal class. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A(++) _V )  =  ( _V  \  A )
 
Theoremsymdifid 23778 Symmetric difference yields the empty class with the same argument twice. (Contributed by Scott Fenton, 25-Apr-2012.)
 |-  ( A(++) A )  =  (/)
 
Theoremsymdifass 23779 Symmetric difference associates. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A(++) ( B(++) C ) )  =  ( ( A(++) B )(++) C )
 
Theorembrsymdif 23780 The binary relationship of a symmetric difference. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  ( A ( R(++) S ) B  <->  -.  ( A R B 
 <->  A S B ) )
 
18.7.28  Quantifier-free definitions
 
Syntaxctxp 23781 Declare the syntax for tail cross product.
 class  ( A 
 (x)  B )
 
Syntaxcpprod 23782 Declare the syntax for the parallel product.
 class pprod ( R ,  S )
 
Syntaxcsset 23783 Declare the subset relationship class.
 class  SSet
 
Syntaxctrans 23784 Declare the transitive set class.
 class  Trans
 
Syntaxcbigcup 23785 Declare the set union relationship.
 class  Bigcup
 
Syntaxcfix 23786 Declare the syntax for the fixpoints of a class.
 class  Fix A
 
Syntaxclimits 23787 Declare the class of limit ordinals.
 class  Limits
 
Syntaxcfuns 23788 Declare the syntax for the class of all function.
 class  Funs
 
Syntaxcsingle 23789 Declare the syntax for the singleton function.
 class Singleton
 
Syntaxcsingles 23790 Declare the syntax for the class of all singletons.
 class  Singletons
 
Syntaxcimage 23791 Declare the syntax for the image functor.
 class Image A
 
Syntaxccart 23792 Declare the syntax for the cartesian function.
 class Cart
 
Syntaxcimg 23793 Declare the syntax for the image function.
 class Img
 
Syntaxcdomain 23794 Declare the syntax for the domain function.
 class Domain
 
Syntaxcrange 23795 Declare the syntax for the range function.
 class Range
 
Syntaxcapply 23796 Declare the syntax for the application function.
 class Apply
 
Syntaxccup 23797 Declare the syntax for the cup function.
 class Cup
 
Syntaxccap 23798 Declare the syntax for the cap function.
 class Cap
 
Syntaxcsuccf 23799 Declare the syntax for the successor function.
 class Succ
 
Syntaxcfunpart 23800 Declare the syntax for the functional part functor.
 class Funpart F
    < 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 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31421
  Copyright terms: Public domain < Previous  Next >