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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfixssdm 25701 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A 
 C_  dom  A
 
Theoremfixssrn 25702 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A 
 C_  ran  A
 
Theoremfixcnv 25703 The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A  =  Fix `' A
 
Theoremfixun 25704 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix ( A  u.  B )  =  ( Fix A  u.  Fix B )
 
Theoremellimits 25705 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A  e.  _V   =>    |-  ( A  e.  Limits  <->  Lim  A )
 
Theoremlimitssson 25706 The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Limits  C_  On
 
Theoremdfom5b 25707 A quantifier-free definition of 
om that does not depend on ax-inf 7582. (Note: label was changed from dfom5 7594 to dfom5b 25707 to prevent naming conflict. NM 12-Feb-2013) (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  om  =  ( On  i^i  |^| Limits )
 
Theoremsscoid 25708 A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
 |-  ( A  C_  (  _I  o.  B )  <->  ( Rel  A  /\  A  C_  B )
 )
 
Theoremdffun10 25709 Another potential definition of functionhood. Based on statements in http://people.math.gatech.edu/~belinfan/research/autoreas/otter/sum/fs/. (Contributed by Scott Fenton, 30-Aug-2017.)
 |-  ( Fun  F  <->  F  C_  (  _I 
 o.  ( _V  \  (
 ( _V  \  _I  )  o.  F ) ) ) )
 
Theoremelfuns 25710 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  F  e.  _V   =>    |-  ( F  e.  Funs  <->  Fun  F )
 
Theoremelfunsg 25711 Closed form of elfuns 25710. (Contributed by Scott Fenton, 2-May-2014.)
 |-  ( F  e.  V  ->  ( F  e.  Funs  <->  Fun  F ) )
 
Theorembrsingle 25712 The binary relationship form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ASingleton B  <->  B  =  { A } )
 
Theoremelsingles 25713* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( A  e.  Singletons 
 <-> 
 E. x  A  =  { x } )
 
Theoremfnsingle 25714 The singleton relationship is a function over the universe. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- Singleton  Fn  _V
 
Theoremfvsingle 25715 The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.)
 |-  (Singleton `  A )  =  { A }
 
Theoremdfsingles2 25716* Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Singletons  =  { x  |  E. y  x  =  { y } }
 
Theoremsnelsingles 25717 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  A  e.  _V   =>    |- 
 { A }  e.  Singletons
 
Theoremdfiota3 25718 A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( iota x ph )  = 
 U. U. ( { { x  |  ph } }  i^i 
 Singletons )
 
Theoremdffv5 25719 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( F `  A )  = 
 U. U. ( { ( F " { A }
 ) }  i^i  Singletons )
 
Theoremunisnif 25720 Express union of singleton in terms of  if. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  U. { A }  =  if ( A  e.  _V ,  A ,  (/) )
 
Theorembrimage 25721 Binary relationship form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( AImage R B  <->  B  =  ( R " A ) )
 
Theorembrimageg 25722 Closed form of brimage 25721. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( AImage R B  <->  B  =  ( R " A ) ) )
 
Theoremfunimage 25723 Image A is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun Image A
 
Theoremfnimage 25724* Image R is a function over the set-like portion of  R. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- Image R  Fn  { x  |  ( R
 " x )  e. 
 _V }
 
Theoremimageval 25725* The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
 
Theoremfvimage 25726 The value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  V  /\  ( R " A )  e.  W )  ->  (Image R `  A )  =  ( R " A ) )
 
Theorembrcart 25727 Binary relationship form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Cart C  <->  C  =  ( A  X.  B ) )
 
Theorembrdomain 25728 The binary relationship form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ADomain B  <->  B  =  dom  A )
 
Theorembrrange 25729 The binary relationship form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ARange B  <->  B  =  ran  A )
 
Theorembrdomaing 25730 Closed form of brdomain 25728. (Contributed by Scott Fenton, 2-May-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ADomain B  <->  B  =  dom  A ) )
 
Theorembrrangeg 25731 Closed form of brrange 25729. (Contributed by Scott Fenton, 3-May-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ARange B  <->  B  =  ran  A ) )
 
Theorembrimg 25732 The binary relationship form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Img C  <->  C  =  ( A " B ) )
 
Theorembrapply 25733 The binary relationship form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Apply C  <->  C  =  ( A `  B ) )
 
Theorembrcup 25734 Binary relationship form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Cup C  <->  C  =  ( A  u.  B ) )
 
Theorembrcap 25735 Binary relationship form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Cap C  <->  C  =  ( A  i^i  B ) )
 
Theorembrsuccf 25736 Binary relationship form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ASucc B  <->  B  =  suc  A )
 
Theoremfunpartlem 25737* Lemma for funpartfun 25738. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  ( A  e.  dom  ( (Image
 F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. x ( F
 " { A }
 )  =  { x } )
 
Theoremfunpartfun 25738 The functional part of  F is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun Funpart F
 
Theoremfunpartss 25739 The functional part of  F is a subset of  F. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- Funpart F  C_  F
 
Theoremfunpartfv 25740 The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (Funpart F `
  A )  =  ( F `  A )
 
Theoremfullfunfnv 25741 The full functional part of  F is a function over  _V. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- FullFun F  Fn  _V
 
Theoremfullfunfv 25742 The function value of the full function of  F agrees with  F. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (FullFun F `
  A )  =  ( F `  A )
 
Theorembrfullfun 25743 A binary relationship form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( AFullFun F B  <->  B  =  ( F `  A ) )
 
Theorembrrestrict 25744 The binary relationship form of the Restrict function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Restrict
 C 
 <->  C  =  ( A  |`  B ) )
 
Theoremdfrdg4 25745 A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  rec ( F ,  A )  =  U. ( (
 Funs  i^i  ( `'Domain " On ) )  \  dom  (
 ( `'  _E  o. Domain ) 
 \  Fix ( `'Apply  o.  (
 ( ( _V  X.  { (/) } )  X.  { U. { A } }
 )  u.  ( ( ( Bigcup  o. Img )  |`  ( _V 
 X.  Limits ) )  u.  ( (FullFun F  o.  (Apply  o. pprod (  _I  ,  Bigcup ) ) )  |`  ( _V  X.  ran Succ ) ) ) ) ) ) )
 
Theoremtfrqfree 25746* Calculate a quantifier-free version of the function from tfr1 6649 through tfr3 6651. (Contributed by Scott Fenton, 29-Apr-2014.)
 |-  (
 ( Funs  i^i  ( `'Domain " On ) )  \  dom  ( ( `'  _E  o. Domain )  \  Fix ( `'Apply  o.  (FullFun G  o. Restrict ) ) ) )  =  { f  |  E. x  e.  On  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
 
Theoremdfint3 25747 Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
 |-  |^| A  =  ( _V  \  ( `' ( _V  \  _E  ) " A ) )
 
Theoremimagesset 25748 The Image functor applied to the converse of the subset relationship yields a subset of the subset relationship. (Contributed by Scott Fenton, 14-Apr-2018.)
 |- Image `' SSet  C_ 
 SSet
 
19.7.38  Alternate ordered pairs
 
Syntaxcaltop 25749 Declare the syntax for an alternate ordered pair.
 class  << A ,  B >>
 
Syntaxcaltxp 25750 Declare the syntax for an alternate cross product.
 class  ( A 
 XX.  B )
 
Definitiondf-altop 25751 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 25762), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  << A ,  B >>  =  { { A } ,  { A ,  { B } } }
 
Definitiondf-altxp 25752* Define cross products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.)
 |-  ( A  XX.  B )  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  << x ,  y >> }
 
Theoremaltopex 25753 Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  << A ,  B >>  e.  _V
 
Theoremaltopthsn 25754 Two alternate ordered pairs are equal iff the singletons of their respective elements are equal. Note that this holds regardless of sethood of any of the elements. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  ( <<
 A ,  B >>  = 
 << C ,  D >>  <->  ( { A }  =  { C }  /\  { B }  =  { D } )
 )
 
Theoremaltopeq12 25755 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  (
 ( A  =  B  /\  C  =  D ) 
 ->  << A ,  C >> 
 =  << B ,  D >> )
 
Theoremaltopeq1 25756 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( A  =  B  ->  << A ,  C >>  =  << B ,  C >> )
 
Theoremaltopeq2 25757 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( A  =  B  ->  << C ,  A >>  =  << C ,  B >> )
 
Theoremaltopth1 25758 Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( A  e.  V  ->  (
 << A ,  B >>  = 
 << C ,  D >>  ->  A  =  C )
 )
 
Theoremaltopth2 25759 Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( B  e.  V  ->  (
 << A ,  B >>  = 
 << C ,  D >>  ->  B  =  D )
 )
 
Theoremaltopthg 25760 Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( << A ,  B >> 
 =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
 ) )
 
Theoremaltopthbg 25761 Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)
 |-  (
 ( A  e.  V  /\  D  e.  W ) 
 ->  ( << A ,  B >> 
 =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
 ) )
 
Theoremaltopth 25762 The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that  C and  D are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 4427), requires  D to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <<
 A ,  B >>  = 
 << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremaltopthb 25763 Alternate ordered pair theorem with different sethood requirements. See altopth 25762 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
 |-  A  e.  _V   &    |-  D  e.  _V   =>    |-  ( <<
 A ,  B >>  = 
 << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremaltopthc 25764 Alternate ordered pair theorem with different sethood requirements. See altopth 25762 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <<
 A ,  B >>  = 
 << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremaltopthd 25765 Alternate ordered pair theorem with different sethood requirements. See altopth 25762 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
 |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( <<
 A ,  B >>  = 
 << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremaltxpeq1 25766 Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)
 |-  ( A  =  B  ->  ( A  XX.  C )  =  ( B  XX.  C ) )
 
Theoremaltxpeq2 25767 Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)
 |-  ( A  =  B  ->  ( C  XX.  A )  =  ( C  XX.  B ) )
 
Theoremelaltxp 25768* Membership in alternate cross products. (Contributed by Scott Fenton, 23-Mar-2012.)
 |-  ( X  e.  ( A  XX. 
 B )  <->  E. x  e.  A  E. y  e.  B  X  =  << x ,  y >> )
 
Theoremaltopelaltxp 25769 Alternate ordered pair membership in a cross product. Note that, unlike opelxp 4899, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( <<
 X ,  Y >>  e.  ( A  XX.  B )  <-> 
 ( X  e.  A  /\  Y  e.  B ) )
 
Theoremaltxpsspw 25770 An inclusion rule for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)
 |-  ( A  XX.  B )  C_  ~P
 ~P ( A  u.  ~P B )
 
Theoremaltxpexg 25771 The alternate cross product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A  XX.  B )  e.  _V )
 
Theoremrankaltopb 25772 Compute the rank of an alternate ordered pair. (Contributed by Scott Fenton, 18-Dec-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) ) 
 ->  ( rank `  << A ,  B >> )  =  suc  suc  ( ( rank `  A )  u.  suc  ( rank `  B ) ) )
 
Theoremnfaltop 25773 Bound-variable hypothesis builder for alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x << A ,  B >>
 
Theoremsbcaltop 25774* Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
 |-  ( A  e.  _V  ->  [_ A  /  x ]_ << C ,  D >>  =  << [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >> )
 
19.7.39  Tarskian geometry
 
Syntaxcee 25775 Declare the syntax for the Euclidean space generator.
 class  EE
 
Syntaxcbtwn 25776 Declare the syntax for the Euclidean betweenness predicate.
 class  Btwn
 
Syntaxccgr 25777 Declare the syntax for the Euclidean congruence predicate.
 class Cgr
 
Definitiondf-ee 25778 Define the Euclidean space generator. For details, see elee 25781. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  ( 1
 ... n ) ) )
 
Definitiondf-btwn 25779* Define the Euclidean betweenness predicate. For details, see brbtwn 25786. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  Btwn  =  `' { <. <. x ,  z >. ,  y >.  |  E. n  e.  NN  (
 ( x  e.  ( EE `  n )  /\  z  e.  ( EE `  n )  /\  y  e.  ( EE `  n ) )  /\  E. t  e.  ( 0 [,] 1
 ) A. i  e.  (
 1 ... n ) ( y `  i )  =  ( ( ( 1  -  t )  x.  ( x `  i ) )  +  ( t  x.  (
 z `  i )
 ) ) ) }
 
Definitiondf-cgr 25780* Define the Euclidean congruence predicate. For details, see brcgr 25787. (Contributed by Scott Fenton, 3-Jun-2013.)
 |- Cgr  =  { <. x ,  y >.  | 
 E. n  e.  NN  ( ( x  e.  ( ( EE `  n )  X.  ( EE `  n ) ) 
 /\  y  e.  (
 ( EE `  n )  X.  ( EE `  n ) ) ) 
 /\  sum_ i  e.  (
 1 ... n ) ( ( ( ( 1st `  x ) `  i
 )  -  ( ( 2nd `  x ) `  i ) ) ^
 2 )  =  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  y
 ) `  i )  -  ( ( 2nd `  y
 ) `  i )
 ) ^ 2 ) ) }
 
Theoremelee 25781 Membership in a Euclidean space. We define Euclidean space here using Cartesian coordinates over 
N space. We later abstract away from this using Tarski's geometry axioms, so this exact definition is unimportant. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  ( N  e.  NN  ->  ( A  e.  ( EE
 `  N )  <->  A : ( 1
 ... N ) --> RR )
 )
 
Theoremmptelee 25782* A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.)
 |-  ( N  e.  NN  ->  ( ( k  e.  (
 1 ... N )  |->  ( A F B ) )  e.  ( EE
 `  N )  <->  A. k  e.  (
 1 ... N ) ( A F B )  e.  RR ) )
 
Theoremeleenn 25783 If  A is in  ( EE
`  N ), then  N is a natural. (Contributed by Scott Fenton, 1-Jul-2013.)
 |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )
 
Theoremeleei 25784 The forward direction of elee 25781. (Contributed by Scott Fenton, 1-Jul-2013.)
 |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ...
 N ) --> RR )
 
Theoremeedimeq 25785 A point belongs to at most one Euclidean space. (Contributed by Scott Fenton, 1-Jul-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  A  e.  ( EE `  M ) )  ->  N  =  M )
 
Theorembrbtwn 25786* The binary relationship form of the betweenness predicate. The statement  A  Btwn  <. B ,  C >. should be informally read as " A lies on a line segment between  B and  C. This exact definition is abstracted away by Tarski's geometry axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( A 
 Btwn  <. B ,  C >.  <->  E. t  e.  (
 0 [,] 1 ) A. i  e.  ( 1 ... N ) ( A `
  i )  =  ( ( ( 1  -  t )  x.  ( B `  i
 ) )  +  (
 t  x.  ( C `
  i ) ) ) ) )
 
Theorembrcgr 25787* The binary relationship form of the congruence predicate. The statement  <. A ,  B >.Cgr <. C ,  D >. should be read informally as "the  N dimensional point  A is as far from  B as  C is from  D, or "the line segment  A B is congruent to the line segment  C D. This particular definition is encapsulated by Tarski's axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.Cgr <. C ,  D >. 
 <-> 
 sum_ i  e.  (
 1 ... N ) ( ( ( A `  i )  -  ( B `  i ) ) ^ 2 )  = 
 sum_ i  e.  (
 1 ... N ) ( ( ( C `  i )  -  ( D `  i ) ) ^ 2 ) ) )
 
Theoremfveere 25788 The function value of a point is a real. (Contributed by Scott Fenton, 10-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  I  e.  ( 1 ... N ) )  ->  ( A `  I )  e.  RR )
 
Theoremfveecn 25789 The function value of a point is a complex. (Contributed by Scott Fenton, 10-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  I  e.  ( 1 ... N ) )  ->  ( A `  I )  e.  CC )
 
Theoremeqeefv 25790* Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  ( A  =  B  <->  A. i  e.  ( 1
 ... N ) ( A `  i )  =  ( B `  i ) ) )
 
Theoremeqeelen 25791* Two points are equal iff the square of the distance between them is zero. (Contributed by Scott Fenton, 10-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  ( A  =  B  <->  sum_
 i  e.  ( 1
 ... N ) ( ( ( A `  i )  -  ( B `  i ) ) ^ 2 )  =  0 ) )
 
Theorembrbtwn2 25792* Alternate characterization of betweenness, with no existential quantifiers. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( A 
 Btwn  <. B ,  C >.  <-> 
 ( A. i  e.  (
 1 ... N ) ( ( ( B `  i )  -  ( A `  i ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) )  <_  0  /\  A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( B `  i
 )  -  ( A `
  i ) )  x.  ( ( C `
  j )  -  ( A `  j ) ) )  =  ( ( ( B `  j )  -  ( A `  j ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) ) ) ) )
 
Theoremcolinearalglem1 25793 Lemma for colinearalg 25797. Expand out a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC )
 )  ->  ( (
 ( B  -  A )  x.  ( F  -  D ) )  =  ( ( E  -  D )  x.  ( C  -  A ) )  <-> 
 ( ( B  x.  F )  -  (
 ( A  x.  F )  +  ( B  x.  D ) ) )  =  ( ( C  x.  E )  -  ( ( A  x.  E )  +  ( C  x.  D ) ) ) ) )
 
Theoremcolinearalglem2 25794* Lemma for colinearalg 25797. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( A. i  e.  ( 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( B `  i )  -  ( A `  i ) )  x.  ( ( C `  j )  -  ( A `  j ) ) )  =  ( ( ( B `  j
 )  -  ( A `
  j ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) )  <->  A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( C `  i
 )  -  ( B `
  i ) )  x.  ( ( A `
  j )  -  ( B `  j ) ) )  =  ( ( ( C `  j )  -  ( B `  j ) )  x.  ( ( A `
  i )  -  ( B `  i ) ) ) ) )
 
Theoremcolinearalglem3 25795* Lemma for colinearalg 25797. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( A. i  e.  ( 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( B `  i )  -  ( A `  i ) )  x.  ( ( C `  j )  -  ( A `  j ) ) )  =  ( ( ( B `  j
 )  -  ( A `
  j ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) )  <->  A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( A `  i
 )  -  ( C `
  i ) )  x.  ( ( B `
  j )  -  ( C `  j ) ) )  =  ( ( ( A `  j )  -  ( C `  j ) )  x.  ( ( B `
  i )  -  ( C `  i ) ) ) ) )
 
Theoremcolinearalglem4 25796* Lemma for colinearalg 25797. Prove a disjunction that will be needed in the final proof. (Contributed by Scott Fenton, 27-Jun-2013.)
 |-  (
 ( ( A  e.  ( EE `  N ) 
 /\  C  e.  ( EE `  N ) ) 
 /\  K  e.  RR )  ->  ( A. i  e.  ( 1 ... N ) ( ( ( ( K  x.  (
 ( C `  i
 )  -  ( A `
  i ) ) )  +  ( A `
  i ) )  -  ( A `  i ) )  x.  ( ( C `  i )  -  ( A `  i ) ) )  <_  0  \/  A. i  e.  ( 1
 ... N ) ( ( ( C `  i )  -  (
 ( K  x.  (
 ( C `  i
 )  -  ( A `
  i ) ) )  +  ( A `
  i ) ) )  x.  ( ( A `  i )  -  ( ( K  x.  ( ( C `
  i )  -  ( A `  i ) ) )  +  ( A `  i ) ) ) )  <_  0  \/  A. i  e.  (
 1 ... N ) ( ( ( A `  i )  -  ( C `  i ) )  x.  ( ( ( K  x.  ( ( C `  i )  -  ( A `  i ) ) )  +  ( A `  i ) )  -  ( C `  i ) ) )  <_  0
 ) )
 
Theoremcolinearalg 25797* An algebraic characterization of colinearity. Note the similarity to brbtwn2 25792. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. )  <->  A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( B `  i
 )  -  ( A `
  i ) )  x.  ( ( C `
  j )  -  ( A `  j ) ) )  =  ( ( ( B `  j )  -  ( A `  j ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) ) ) )
 
Theoremeleesub 25798* Membership of a subtraction mapping in a Euclidean space. (Contributed by Scott Fenton, 17-Jul-2013.)
 |-  C  =  ( i  e.  (
 1 ... N )  |->  ( ( A `  i
 )  -  ( B `
  i ) ) )   =>    |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 ->  C  e.  ( EE
 `  N ) )
 
Theoremeleesubd 25799* Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 25798. (Contributed by Scott Fenton, 17-Jul-2013.)
 |-  ( ph  ->  C  =  ( i  e.  ( 1
 ... N )  |->  ( ( A `  i
 )  -  ( B `
  i ) ) ) )   =>    |-  ( ( ph  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 ->  C  e.  ( EE
 `  N ) )
 
19.7.40  Tarski's axioms for geometry
 
Theoremaxdimuniq 25800 The unique dimensional axiom. If a point is in  N dimensional space and in  M dimensional space, then  N  =  M. This axiom is not traditionally presented with Tarski's axioms, but we require it here as we are considering spaces in arbitrary dimensions. (Contributed by Scott Fenton, 24-Sep-2013.)
 |-  (
 ( ( N  e.  NN  /\  A  e.  ( EE `  N ) ) 
 /\  ( M  e.  NN  /\  A  e.  ( EE `  M ) ) )  ->  N  =  M )
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