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Theorem List for Metamath Proof Explorer - 24501-24600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaltopthg 24501 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 24502 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 24503 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 4245), 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 24504 Alternate ordered pair theorem with different sethood requirements. See altopth 24503 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 24505 Alternate ordered pair theorem with different sethood requirements. See altopth 24503 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 24506 Alternate ordered pair theorem with different sethood requirements. See altopth 24503 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 24507 Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)
 |-  ( A  =  B  ->  ( A  XX.  C )  =  ( B  XX.  C ) )
 
Theoremaltxpeq2 24508 Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)
 |-  ( A  =  B  ->  ( C  XX.  A )  =  ( C  XX.  B ) )
 
Theoremelaltxp 24509* 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 24510 Alternate ordered pair membership in a cross product. Note that, unlike opelxp 4719, 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 24511 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 24512 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 24513 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 24514 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 24515* 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 >> )
 
18.7.32  Tarskian geometry
 
Syntaxcee 24516 Declare the syntax for the Euclidean space generator.
 class  EE
 
Syntaxcbtwn 24517 Declare the syntax for the Euclidean betweenness predicate.
 class  Btwn
 
Syntaxccgr 24518 Declare the syntax for the Euclidean congruence predicate.
 class Cgr
 
Definitiondf-ee 24519 Define the Euclidean space generator. For details, see elee 24522. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  ( 1
 ... n ) ) )
 
Definitiondf-btwn 24520* Define the Euclidean betweenness predicate. For details, see brbtwn 24527. (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 24521* Define the Euclidean congruence predicate. For details, see brcgr 24528. (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 24522 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 24523* 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 24524 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 24525 The forward direction of elee 24522. (Contributed by Scott Fenton, 1-Jul-2013.)
 |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ...
 N ) --> RR )
 
Theoremeedimeq 24526 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 24527* 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 24528* 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 24529 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 24530 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 24531* 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 24532* 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 24533* 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 24534 Lemma for colinearalg 24538. 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 24535* Lemma for colinearalg 24538. 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 24536* Lemma for colinearalg 24538. 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 24537* Lemma for colinearalg 24538. 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 24538* An algebraic characterization of colinearity. Note the similarity to brbtwn2 24533. (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 24539* 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 24540* Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 24539. (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 ) )
 
18.7.33  Tarski's axioms for geometry
 
Theoremaxdimuniq 24541 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 24542  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 24543 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 24544 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 24545* Lemma for axsegcon 24555. 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 24546* Lemma for axsegcon 24555. 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 24547* Lemma for axsegcon 24555. 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 24548* Lemma for axsegcon 24555. 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 24549* Lemma for axsegcon 24555. 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 24550* Lemma for axsegcon 24555. 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 24551* Lemma for axsegcon 24555. 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 24552* Lemma for axsegcon 24555. 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 24553* Lemma for axsegcon 24555. 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 24554* Lemma for axsegcon 24555. Show that the scaling constant from axsegconlem7 24551 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 24555* 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 24556* Lemma for ax5seg 24566. 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 24557* Lemma for ax5seg 24566. 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 24558 Lemma for ax5seg 24566. (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 24559* Lemma for ax5seg 24566. 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 24560* Lemma for ax5seg 24566. 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 24561* Lemma for ax5seg 24566. 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 24562* Lemma for ax5seg 24566. 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 24563 Lemma for ax5seg 24566. 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 24564 Lemma for ax5seg 24566. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 24563. (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 24565* Lemma for ax5seg 24566. Take the calculation in ax5seglem8 24564 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 24566 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 24567 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 24568* Lemma for axpasch 24569. 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 24569* 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 24570 Lemma for axlowdim 24589. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  (
 ( 3 ... N )  X.  { 0 } ) : ( 3
 ... N ) --> RR
 
Theoremaxlowdimlem2 24571 Lemma for axlowdim 24589. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  (
 ( 1 ... 2
 )  i^i  ( 3 ... N ) )  =  (/)
 
Theoremaxlowdimlem3 24572 Lemma for axlowdim 24589. 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 24573 Lemma for axlowdim 24589. 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 24574 Lemma for axlowdim 24589. 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 24575 Lemma for axlowdim 24589. 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 24576 Lemma for axlowdim 24589. 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 24577 Lemma for axlowdim 24589. 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 24578 Lemma for axlowdim 24589. 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 24579 Lemma for axlowdim 24589. 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 24580 Lemma for axlowdim 24589. 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 24581 Lemma for axlowdim 24589. 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 24582 Lemma for axlowdim 24589. 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 24583 Lemma for axlowdim 24589. Take two possible  Q from axlowdimlem10 24579. 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 24584* Lemma for axlowdim 24589. 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 24585* Lemma for axlowdim 24589. 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 24586 Lemma for axlowdim 24589. 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 24587* 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 24588. (Contributed by Scott Fenton, 22-Apr-2013.)
 |-  ( N  e.  NN  ->  E. x  e.  ( EE
 `  N ) E. y  e.  ( EE `  N ) x  =/=  y )
 
Theoremaxlowdim2 24588* 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 24589* 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 24590* Lemma for axeuclid 24591. 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 24591* 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 >. ) ) )
 
Theoremaxcontlem1 24592* Lemma for axcont 24604. Change bound variables for later use. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( x `  i )  =  (
 ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i
 ) ) ) ) ) }   =>    |-  F  =  { <. y ,  s >.  |  ( y  e.  D  /\  ( s  e.  (
 0 [,)  +oo )  /\  A. j  e.  ( 1
 ... N ) ( y `  j )  =  ( ( ( 1  -  s )  x.  ( Z `  j ) )  +  ( s  x.  ( U `  j ) ) ) ) ) }
 
Theoremaxcontlem2 24593* Lemma for axcont 24604. The idea here is to set up a mapping  F that will allow us to transfer dedekind 24082 to two sets of points. Here, we set up  F and show its domain and range. (Contributed by Scott Fenton, 17-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) )  /\  Z  =/=  U )  ->  F : D -1-1-onto-> ( 0 [,)  +oo ) )
 
Theoremaxcontlem3 24594* Lemma for axcont 24604. Given the separation assumption,  B is a subset of  D. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   =>    |-  ( ( ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N ) 
 /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/=  U ) )  ->  B  C_  D )
 
Theoremaxcontlem4 24595* Lemma for axcont 24604. Given the separation assumption,  A is a subset of  D. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   =>    |-  ( ( ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N ) 
 /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  (
 ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  B  =/=  (/) )  /\  Z  =/=  U ) )  ->  A  C_  D )
 
Theoremaxcontlem5 24596* Lemma for axcont 24604. Compute the value of  F. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) ) 
 /\  Z  =/=  U )  /\  P  e.  D )  ->  ( ( F `
  P )  =  T  <->  ( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  (
 ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i ) ) ) ) ) )
 
Theoremaxcontlem6 24597* Lemma for axcont 24604. State the defining properties of the value of  F (Contributed by Scott Fenton, 19-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) ) 
 /\  Z  =/=  U )  /\  P  e.  D )  ->  ( ( F `
  P )  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  (
 ( ( 1  -  ( F `  P ) )  x.  ( Z `
  i ) )  +  ( ( F `
  P )  x.  ( U `  i
 ) ) ) ) )
 
Theoremaxcontlem7 24598* Lemma for axcont 24604. Given two points in  D, one preceeds the other iff its scaling constant is less than the other point's. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) ) 
 /\  Z  =/=  U )  /\  ( P  e.  D  /\  Q  e.  D ) )  ->  ( P 
 Btwn  <. Z ,  Q >.  <-> 
 ( F `  P )  <_  ( F `  Q ) ) )
 
Theoremaxcontlem8 24599* Lemma for axcont 24604. A point in  D is between two others if its function value falls in the middle. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) ) 
 /\  Z  =/=  U )  /\  ( P  e.  D  /\  Q  e.  D  /\  R  e.  D ) )  ->  ( (
 ( F `  P )  <_  ( F `  Q )  /\  ( F `
  Q )  <_  ( F `  R ) )  ->  Q  Btwn  <. P ,  R >. ) )
 
Theoremaxcontlem9 24600* Lemma for axcont 24604. Given the separation assumption, all values of  F over  A are less than or equal to all values of  F over  B. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( N  e.  NN  /\  ( A  C_  ( EE `  N ) 
 /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  (
 ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  B  =/=  (/) )  /\  Z  =/=  U ) )  ->  A. n  e.  ( F " A ) A. m  e.  ( F " B ) n  <_  m )
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