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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-altop 23901 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 23912), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  << A ,  B >>  =  { { A } ,  { A ,  { B } } }
 
Definitiondf-altxp 23902* 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 23903 Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  << A ,  B >>  e.  _V
 
Theoremaltopthsn 23904 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 23905 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  (
 ( A  =  B  /\  C  =  D ) 
 ->  << A ,  C >> 
 =  << B ,  D >> )
 
Theoremaltopeq1 23906 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( A  =  B  ->  << A ,  C >>  =  << B ,  C >> )
 
Theoremaltopeq2 23907 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( A  =  B  ->  << C ,  A >>  =  << C ,  B >> )
 
Theoremaltopth1 23908 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 23909 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 23910 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 23911 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 23912 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 4243), 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 23913 Alternate ordered pair theorem with different sethood requirements. See altopth 23912 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 23914 Alternate ordered pair theorem with different sethood requirements. See altopth 23912 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 23915 Alternate ordered pair theorem with different sethood requirements. See altopth 23912 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 23916 Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)
 |-  ( A  =  B  ->  ( A  XX.  C )  =  ( B  XX.  C ) )
 
Theoremaltxpeq2 23917 Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)
 |-  ( A  =  B  ->  ( C  XX.  A )  =  ( C  XX.  B ) )
 
Theoremelaltxp 23918* 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 23919 Alternate ordered pair membership in a cross product. Note that, unlike opelxp 4717, 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 23920 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 23921 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 23922 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 23923 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 23924* 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 >> )
 
18.7.30  Tarskian geometry
 
Syntaxcee 23925 Declare the syntax for the Euclidean space generator.
 class  EE
 
Syntaxcbtwn 23926 Declare the syntax for the Euclidean betweenness predicate.
 class  Btwn
 
Syntaxccgr 23927 Declare the syntax for the Euclidean congruence predicate.
 class Cgr
 
Definitiondf-ee 23928 Define the Euclidean space generator. For details, see elee 23931. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  ( 1
 ... n ) ) )
 
Definitiondf-btwn 23929* Define the Euclidean betweenness predicate. For details, see brbtwn 23936. (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 23930* Define the Euclidean congruence predicate. For details, see brcgr 23937. (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 23931 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 23932* 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 23933 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 23934 The forward direction of elee 23931. (Contributed by Scott Fenton, 1-Jul-2013.)
 |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ...
 N ) --> RR )
 
Theoremeedimeq 23935 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 23936* 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 23937* 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 23938 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 23939 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 23940* 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 23941* 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 23942* 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 23943 Lemma for colinearalg 23947. 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 23944* Lemma for colinearalg 23947. 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 23945* Lemma for colinearalg 23947. 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 23946* Lemma for colinearalg 23947. 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 23947* An algebraic characterization of colinearity. Note the similarity to brbtwn2 23942. (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 23948* 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 23949* Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 23948. (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 ) )
 
18.7.31  Tarski's axioms for geometry
 
Theoremaxdimuniq 23950 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 )
 
Theoremaxcgrrflx 23951  A is as far from  B as  B is from  A. Axiom A1 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.Cgr <. B ,  A >. )
 
Theoremaxcgrtr 23952 Congruence is transitive. Axiom A2 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) 
 /\  F  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.Cgr <. C ,  D >.  /\ 
 <. A ,  B >.Cgr <. E ,  F >. ) 
 ->  <. C ,  D >.Cgr
 <. E ,  F >. ) )
 
Theoremaxcgrid 23953 If there is no distance between  A and  B, then  A  =  B. Axiom A3 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.Cgr
 <. C ,  C >.  ->  A  =  B )
 )
 
Theoremaxsegconlem1 23954* Lemma for axsegcon 23964. Handle the degenerate case. (Contributed by Scott Fenton, 7-Jun-2013.)
 |-  (
 ( A  =  B  /\  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) ) )  ->  E. x  e.  ( EE `  N ) E. t  e.  (
 0 [,] 1 ) (
 A. i  e.  (
 1 ... N ) ( B `  i )  =  ( ( ( 1  -  t )  x.  ( A `  i ) )  +  ( t  x.  ( x `  i ) ) )  /\  sum_ i  e.  ( 1 ... N ) ( ( ( B `  i )  -  ( x `  i ) ) ^
 2 )  =  sum_ i  e.  ( 1 ...
 N ) ( ( ( C `  i
 )  -  ( D `
  i ) ) ^ 2 ) ) )
 
Theoremaxsegconlem2 23955* Lemma for axsegcon 23964. Show that the square of the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   =>    |-  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  S  e.  RR )
 
Theoremaxsegconlem3 23956* Lemma for axsegcon 23964. Show that the square of the distance between two points is non-negative. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   =>    |-  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  0  <_  S )
 
Theoremaxsegconlem4 23957* Lemma for axsegcon 23964. Show that the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   =>    |-  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  ( sqr `  S )  e.  RR )
 
Theoremaxsegconlem5 23958* Lemma for axsegcon 23964. Show that the distance between two points is non-negative. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   =>    |-  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  0  <_  ( sqr `  S )
 )
 
Theoremaxsegconlem6 23959* Lemma for axsegcon 23964. Show that the distance between two distinct points is positive. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   =>    |-  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) 
 /\  A  =/=  B )  ->  0  <  ( sqr `  S ) )
 
Theoremaxsegconlem7 23960* Lemma for axsegcon 23964. Show that a particular ratio of distances is in the closed unit interval. (Contributed by Scott Fenton, 18-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   &    |-  T  =  sum_ p  e.  (
 1 ... N ) ( ( ( C `  p )  -  ( D `  p ) ) ^ 2 )   =>    |-  ( ( ( A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( sqr `  S )  /  ( ( sqr `  S )  +  ( sqr `  T ) ) )  e.  ( 0 [,] 1 ) )
 
Theoremaxsegconlem8 23961* Lemma for axsegcon 23964. Show that a particular mapping generates a point. (Contributed by Scott Fenton, 18-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   &    |-  T  =  sum_ p  e.  (
 1 ... N ) ( ( ( C `  p )  -  ( D `  p ) ) ^ 2 )   &    |-  F  =  ( k  e.  (
 1 ... N )  |->  ( ( ( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `  k ) )  -  ( ( sqr `  T )  x.  ( A `  k ) ) )  /  ( sqr `  S ) ) )   =>    |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) 
 /\  A  =/=  B )  /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) )  ->  F  e.  ( EE `  N ) )
 
Theoremaxsegconlem9 23962* Lemma for axsegcon 23964. Show that  B F is congruent to  C D. (Contributed by Scott Fenton, 19-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   &    |-  T  =  sum_ p  e.  (
 1 ... N ) ( ( ( C `  p )  -  ( D `  p ) ) ^ 2 )   &    |-  F  =  ( k  e.  (
 1 ... N )  |->  ( ( ( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `  k ) )  -  ( ( sqr `  T )  x.  ( A `  k ) ) )  /  ( sqr `  S ) ) )   =>    |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) 
 /\  A  =/=  B )  /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) )  ->  sum_ i  e.  ( 1 ... N ) ( ( ( B `  i )  -  ( F `  i ) ) ^
 2 )  =  sum_ i  e.  ( 1 ...
 N ) ( ( ( C `  i
 )  -  ( D `
  i ) ) ^ 2 ) )
 
Theoremaxsegconlem10 23963* Lemma for axsegcon 23964. Show that the scaling constant from axsegconlem7 23960 produces the betweenness condition for  A,  B and  F. (Contributed by Scott Fenton, 21-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   &    |-  T  =  sum_ p  e.  (
 1 ... N ) ( ( ( C `  p )  -  ( D `  p ) ) ^ 2 )   &    |-  F  =  ( k  e.  (
 1 ... N )  |->  ( ( ( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `  k ) )  -  ( ( sqr `  T )  x.  ( A `  k ) ) )  /  ( sqr `  S ) ) )   =>    |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) 
 /\  A  =/=  B )  /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) )  ->  A. i  e.  ( 1 ... N ) ( B `  i )  =  (
 ( ( 1  -  ( ( sqr `  S )  /  ( ( sqr `  S )  +  ( sqr `  T ) ) ) )  x.  ( A `  i ) )  +  ( ( ( sqr `  S )  /  ( ( sqr `  S )  +  ( sqr `  T ) ) )  x.  ( F `  i ) ) ) )
 
Theoremaxsegcon 23964* Any segment  A B can be extended to a point  x such that  B x is congruent to  C D. Axiom A4 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 4-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  E. x  e.  ( EE `  N ) ( B  Btwn  <. A ,  x >.  /\  <. B ,  x >.Cgr <. C ,  D >. ) )
 
Theoremax5seglem1 23965* Lemma for ax5seg 23975. Rexpress a one congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( T  e.  (
 0 [,] 1 )  /\  A. i  e.  ( 1
 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) ) ) )  ->  sum_ j  e.  ( 1
 ... N ) ( ( ( A `  j )  -  ( B `  j ) ) ^ 2 )  =  ( ( T ^
 2 )  x.  sum_ j  e.  ( 1 ...
 N ) ( ( ( A `  j
 )  -  ( C `
  j ) ) ^ 2 ) ) )
 
Theoremax5seglem2 23966* Lemma for ax5seg 23975. Rexpress another congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( T  e.  (
 0 [,] 1 )  /\  A. i  e.  ( 1
 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) ) ) )  ->  sum_ j  e.  ( 1
 ... N ) ( ( ( B `  j )  -  ( C `  j ) ) ^ 2 )  =  ( ( ( 1  -  T ) ^
 2 )  x.  sum_ j  e.  ( 1 ...
 N ) ( ( ( A `  j
 )  -  ( C `
  j ) ) ^ 2 ) ) )
 
Theoremax5seglem3a 23967 Lemma for ax5seg 23975. (Contributed by Scott Fenton, 7-May-2015.)
 |-  (
 ( ( N  e.  NN  /\  ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N ) )  ->  ( ( ( A `
  j )  -  ( C `  j ) )  e.  RR  /\  ( ( D `  j )  -  ( F `  j ) )  e.  RR ) )
 
Theoremax5seglem3 23968* Lemma for ax5seg 23975. Combine congruences for points on a line. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )  /\  ( ( T  e.  ( 0 [,] 1
 )  /\  S  e.  ( 0 [,] 1
 ) )  /\  ( A. i  e.  (
 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  (
 ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i ) ) ) ) ) 
 /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
 <. E ,  F >. ) )  ->  sum_ j  e.  ( 1 ... N ) ( ( ( A `  j )  -  ( C `  j ) ) ^
 2 )  =  sum_ j  e.  ( 1 ...
 N ) ( ( ( D `  j
 )  -  ( F `
  j ) ) ^ 2 ) )
 
Theoremax5seglem4 23969* Lemma for ax5seg 23975. Given two distinct points, the scaling constant in a betweenness statement is non-zero. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) ) 
 /\  A. i  e.  (
 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) )  /\  A  =/=  B )  ->  T  =/=  0 )
 
Theoremax5seglem5 23970* Lemma for ax5seg 23975. If  B is between  A and  C, and  A is distinct from  B, then  A is distinct from  C. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) ) 
 /\  ( A  =/=  B 
 /\  T  e.  (
 0 [,] 1 )  /\  A. i  e.  ( 1
 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) ) ) )  ->  sum_ j  e.  ( 1
 ... N ) ( ( ( A `  j )  -  ( C `  j ) ) ^ 2 )  =/=  0 )
 
Theoremax5seglem6 23971* Lemma for ax5seg 23975. Given two line segments that are divided into pieces, if the pieces are congruent, then the scaling constant is the same. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) 
 /\  C  e.  ( EE `  N ) ) 
 /\  ( D  e.  ( EE `  N ) 
 /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1
 ) )  /\  ( A. i  e.  (
 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  (
 ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i ) ) ) ) ) 
 /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
 <. E ,  F >. ) )  ->  T  =  S )
 
Theoremax5seglem7 23972 Lemma for ax5seg 23975. An algebraic calculation needed further down the line. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  A  e.  CC   &    |-  T  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( T  x.  ( ( C  -  D ) ^
 2 ) )  =  ( ( ( ( ( ( 1  -  T )  x.  A )  +  ( T  x.  C ) )  -  D ) ^ 2
 )  +  ( ( 1  -  T )  x.  ( ( T  x.  ( ( A  -  C ) ^
 2 ) )  -  ( ( A  -  D ) ^ 2
 ) ) ) )
 
Theoremax5seglem8 23973 Lemma for ax5seg 23975. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 23972. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( ( A  e.  CC  /\  T  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( T  x.  ( ( C  -  D ) ^
 2 ) )  =  ( ( ( ( ( ( 1  -  T )  x.  A )  +  ( T  x.  C ) )  -  D ) ^ 2
 )  +  ( ( 1  -  T )  x.  ( ( T  x.  ( ( A  -  C ) ^
 2 ) )  -  ( ( A  -  D ) ^ 2
 ) ) ) ) )
 
Theoremax5seglem9 23974* Lemma for ax5seg 23975. Take the calculation in ax5seglem8 23973 and turn it into a series of measurements. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  (
 ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) )  /\  ( T  e.  (
 0 [,] 1 )  /\  A. i  e.  ( 1
 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) ) ) )  ->  ( T  x.  sum_ j  e.  ( 1 ... N ) ( ( ( C `  j )  -  ( D `  j ) ) ^
 2 ) )  =  ( sum_ j  e.  (
 1 ... N ) ( ( ( B `  j )  -  ( D `  j ) ) ^ 2 )  +  ( ( 1  -  T )  x.  (
 ( T  x.  sum_ j  e.  ( 1 ...
 N ) ( ( ( A `  j
 )  -  ( C `
  j ) ) ^ 2 ) )  -  sum_ j  e.  (
 1 ... N ) ( ( ( A `  j )  -  ( D `  j ) ) ^ 2 ) ) ) ) )
 
Theoremax5seg 23975 The five segment axiom. Take two triangles  A D C and  E H G, a point  B on  A C, and a point  F on  E G. If all corresponding line segments except for  C D and  G H are congruent, then so are  C D and  G H. Axiom A5 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  G  e.  ( EE `  N ) 
 /\  H  e.  ( EE `  N ) ) )  ->  ( (
 ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. ) 
 /\  ( <. A ,  B >.Cgr <. E ,  F >.  /\  <. B ,  C >.Cgr
 <. F ,  G >. ) 
 /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr
 <. F ,  H >. ) )  ->  <. C ,  D >.Cgr <. G ,  H >. ) )
 
Theoremaxbtwnid 23976 Points are indivisible. That is, if  A lies between  B and  B, then  A  =  B. Axiom A6 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  ( A  Btwn  <. B ,  B >.  ->  A  =  B ) )
 
Theoremaxpaschlem 23977* Lemma for axpasch 23978. Set up coefficents used in the proof. (Contributed by Scott Fenton, 5-Jun-2013.)
 |-  (
 ( T  e.  (
 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  ->  E. r  e.  (
 0 [,] 1 ) E. p  e.  ( 0 [,] 1 ) ( p  =  ( ( 1  -  r )  x.  ( 1  -  T ) )  /\  r  =  ( ( 1  -  p )  x.  (
 1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p )  x.  S ) ) )
 
Theoremaxpasch 23978* The inner Pasch axiom. Take a triangle  A C E, a point  D on  A C, and a point  B extending  C E. Then  A E and  D B intersect at some point  x. Axiom A7 of [Schwabhauser] p. 12. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  (
 ( D  Btwn  <. A ,  C >.  /\  E  Btwn  <. B ,  C >. ) 
 ->  E. x  e.  ( EE `  N ) ( x  Btwn  <. D ,  B >.  /\  x  Btwn  <. E ,  A >. ) ) )
 
Theoremaxlowdimlem1 23979 Lemma for axlowdim 23998. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  (
 ( 3 ... N )  X.  { 0 } ) : ( 3
 ... N ) --> RR
 
Theoremaxlowdimlem2 23980 Lemma for axlowdim 23998. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  (
 ( 1 ... 2
 )  i^i  ( 3 ... N ) )  =  (/)
 
Theoremaxlowdimlem3 23981 Lemma for axlowdim 23998. Set up a union property for an interval of integers. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  ( N  e.  ( ZZ>= `  2 )  ->  ( 1
 ... N )  =  ( ( 1 ... 2 )  u.  (
 3 ... N ) ) )
 
Theoremaxlowdimlem4 23982 Lemma for axlowdim 23998. Set up a particular constant function. (Contributed by Scott Fenton, 17-Apr-2013.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  { <. 1 ,  A >. ,  <. 2 ,  B >. } :
 ( 1 ... 2
 ) --> RR
 
Theoremaxlowdimlem5 23983 Lemma for axlowdim 23998. Show that a particular union is a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( N  e.  ( ZZ>= `  2 )  ->  ( { <. 1 ,  A >. , 
 <. 2 ,  B >. }  u.  ( ( 3
 ... N )  X.  { 0 } ) )  e.  ( EE `  N ) )
 
Theoremaxlowdimlem6 23984 Lemma for axlowdim 23998. Show that three points are non-colinear. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  A  =  ( { <. 1 ,  0 >. ,  <. 2 ,  0 >. }  u.  ( ( 3 ...
 N )  X.  {
 0 } ) )   &    |-  B  =  ( { <. 1 ,  1 >. ,  <. 2 ,  0
 >. }  u.  ( ( 3 ... N )  X.  { 0 } ) )   &    |-  C  =  ( { <. 1 ,  0
 >. ,  <. 2 ,  1
 >. }  u.  ( ( 3 ... N )  X.  { 0 } ) )   =>    |-  ( N  e.  ( ZZ>=
 `  2 )  ->  -.  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) )
 
Theoremaxlowdimlem7 23985 Lemma for axlowdim 23998. Set up a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   =>    |-  ( N  e.  ( ZZ>=
 `  3 )  ->  P  e.  ( EE `  N ) )
 
Theoremaxlowdimlem8 23986 Lemma for axlowdim 23998. Calulate the value of  P at three. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   =>    |-  ( P `  3
 )  =  -u 1
 
Theoremaxlowdimlem9 23987 Lemma for axlowdim 23998. Calulate the value of  P away from three. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   =>    |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  3
 )  ->  ( P `  K )  =  0 )
 
Theoremaxlowdimlem10 23988 Lemma for axlowdim 23998. Set up a family of points in Euclidean space. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  (
 ( ( 1 ...
 N )  \  {
 ( I  +  1 ) } )  X.  { 0 } ) )   =>    |-  ( ( N  e.  NN  /\  I  e.  (
 1 ... ( N  -  1 ) ) ) 
 ->  Q  e.  ( EE
 `  N ) )
 
Theoremaxlowdimlem11 23989 Lemma for axlowdim 23998. Calculate the value of  Q at its distinguished point. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  (
 ( ( 1 ...
 N )  \  {
 ( I  +  1 ) } )  X.  { 0 } ) )   =>    |-  ( Q `  ( I  +  1 ) )  =  1
 
Theoremaxlowdimlem12 23990 Lemma for axlowdim 23998. Calculate the value of  Q away from its distunguished point. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  (
 ( ( 1 ...
 N )  \  {
 ( I  +  1 ) } )  X.  { 0 } ) )   =>    |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  ( I  +  1 )
 )  ->  ( Q `  K )  =  0 )
 
Theoremaxlowdimlem13 23991 Lemma for axlowdim 23998. Establish that  P and 
Q are different points. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   &    |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N ) 
 \  { ( I  +  1 ) }
 )  X.  { 0 } ) )   =>    |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  -  1 ) ) )  ->  P  =/=  Q )
 
Theoremaxlowdimlem14 23992 Lemma for axlowdim 23998. Take two possible  Q from axlowdimlem10 23988. They are the same iff their distunguished values are the same. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  (
 ( ( 1 ...
 N )  \  {
 ( I  +  1 ) } )  X.  { 0 } ) )   &    |-  R  =  ( { <. ( J  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N ) 
 \  { ( J  +  1 ) }
 )  X.  { 0 } ) )   =>    |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  -  1 ) ) 
 /\  J  e.  (
 1 ... ( N  -  1 ) ) ) 
 ->  ( Q  =  R  ->  I  =  J ) )
 
Theoremaxlowdimlem15 23993* Lemma for axlowdim 23998. Set up a one to one function of points. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  F  =  ( i  e.  (
 1 ... ( N  -  1 ) )  |->  if ( i  =  1 ,  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 ) ,  ( { <. ( i  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N ) 
 \  { ( i  +  1 ) }
 )  X.  { 0 } ) ) ) )   =>    |-  ( N  e.  ( ZZ>=
 `  3 )  ->  F : ( 1 ... ( N  -  1
 ) ) -1-1-> ( EE
 `  N ) )
 
Theoremaxlowdimlem16 23994* Lemma for axlowdim 23998. Set up a summation that will help establish distance. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   &    |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N ) 
 \  { ( I  +  1 ) }
 )  X.  { 0 } ) )   =>    |-  ( ( N  e.  ( ZZ>= `  3
 )  /\  I  e.  ( 2 ... ( N  -  1 ) ) )  ->  sum_ i  e.  ( 3 ... N ) ( ( P `
  i ) ^
 2 )  =  sum_ i  e.  ( 3 ...
 N ) ( ( Q `  i ) ^ 2 ) )
 
Theoremaxlowdimlem17 23995 Lemma for axlowdim 23998. Establish a congruence result. (Contributed by Scott Fenton, 22-Apr-2013.) (Proof shortened by Mario Carneiro, 22-May-2014.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   &    |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N ) 
 \  { ( I  +  1 ) }
 )  X.  { 0 } ) )   &    |-  A  =  ( { <. 1 ,  X >. ,  <. 2 ,  Y >. }  u.  (
 ( 3 ... N )  X.  { 0 } ) )   &    |-  X  e.  RR   &    |-  Y  e.  RR   =>    |-  ( ( N  e.  ( ZZ>= `  3 )  /\  I  e.  (
 2 ... ( N  -  1 ) ) ) 
 ->  <. P ,  A >.Cgr
 <. Q ,  A >. )
 
Theoremaxlowdim1 23996* The lower dimensional axiom for one dimension. In any dimension, there are at least two distinct points. Theorem 3.13 of [Schwabhauser] p. 32, where it is derived from axlowdim2 23997. (Contributed by Scott Fenton, 22-Apr-2013.)
 |-  ( N  e.  NN  ->  E. x  e.  ( EE
 `  N ) E. y  e.  ( EE `  N ) x  =/=  y )
 
Theoremaxlowdim2 23997* The lower two dimensional axiom. In any space where the dimension is greater than one, there are three non-colinear points. Axiom A8 of [Schwabhauser] p. 12. (Contributed by Scott Fenton, 15-Apr-2013.)
 |-  ( N  e.  ( ZZ>= `  2 )  ->  E. x  e.  ( EE `  N ) E. y  e.  ( EE `  N ) E. z  e.  ( EE `  N )  -.  ( x  Btwn  <. y ,  z >.  \/  y  Btwn  <. z ,  x >.  \/  z  Btwn  <. x ,  y >. ) )
 
Theoremaxlowdim 23998* The general lower dimensional axiom. Take a dimension  N greater than or equal to three. Then, there are three non-colinear points in  N dimensional space that are equidistant from  N  -  1 distinct points. Derived from remarks in "Tarski's System of Geometry", by Alfred Tarski and Steven Givant, Bull. Symbolic Logic Volume 5, Number 2 (1999), 175-214. (Contributed by Scott Fenton, 22-Apr-2013.)
 |-  ( N  e.  ( ZZ>= `  3 )  ->  E. p E. x  e.  ( EE `  N ) E. y  e.  ( EE `  N ) E. z  e.  ( EE `  N ) ( p :
 ( 1 ... ( N  -  1 ) )
 -1-1-> ( EE `  N )  /\  A. i  e.  ( 2 ... ( N  -  1 ) ) ( <. ( p `  1 ) ,  x >.Cgr
 <. ( p `  i
 ) ,  x >.  /\ 
 <. ( p `  1
 ) ,  y >.Cgr <.
 ( p `  i
 ) ,  y >.  /\ 
 <. ( p `  1
 ) ,  z >.Cgr <.
 ( p `  i
 ) ,  z >. ) 
 /\  -.  ( x  Btwn  <. y ,  z >.  \/  y  Btwn  <. z ,  x >.  \/  z  Btwn  <. x ,  y >. ) ) )
 
Theoremaxeuclidlem 23999* Lemma for axeuclid 24000. Handle the algebraic aspects of the theorem. (Contributed by Scott Fenton, 9-Sep-2013.)
 |-  (
 ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  T  e.  ( EE `  N ) ) )  /\  ( P  e.  ( 0 [,] 1 )  /\  Q  e.  ( 0 [,] 1
 )  /\  P  =/=  0 )  /\  A. i  e.  ( 1 ... N ) ( ( ( 1  -  P )  x.  ( A `  i ) )  +  ( P  x.  ( T `  i ) ) )  =  ( ( ( 1  -  Q )  x.  ( B `  i ) )  +  ( Q  x.  ( C `  i ) ) ) )  ->  E. x  e.  ( EE `  N ) E. y  e.  ( EE `  N ) E. r  e.  ( 0 [,] 1 ) E. s  e.  ( 0 [,] 1
 ) E. u  e.  ( 0 [,] 1
 ) A. i  e.  (
 1 ... N ) ( ( B `  i
 )  =  ( ( ( 1  -  r
 )  x.  ( A `
  i ) )  +  ( r  x.  ( x `  i
 ) ) )  /\  ( C `  i )  =  ( ( ( 1  -  s )  x.  ( A `  i ) )  +  ( s  x.  (
 y `  i )
 ) )  /\  ( T `  i )  =  ( ( ( 1  -  u )  x.  ( x `  i
 ) )  +  ( u  x.  ( y `  i ) ) ) ) )
 
Theoremaxeuclid 24000* Euclid's axiom. Take an angle  B A C and a point  D between  B and  C. Now, if you extend the segment  A D to a point  T, then  T lies between two points  x and  y that lie on the angle. Axiom A10 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 9-Sep-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  T  e.  ( EE `  N ) ) )  ->  (
 ( D  Btwn  <. A ,  T >.  /\  D  Btwn  <. B ,  C >.  /\  A  =/=  D ) 
 ->  E. x  e.  ( EE `  N ) E. y  e.  ( EE `  N ) ( B 
 Btwn  <. A ,  x >.  /\  C  Btwn  <. A ,  y >.  /\  T  Btwn  <. x ,  y >. ) ) )
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