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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelee 24601 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 24602* 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 24603 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 24604 The forward direction of elee 24601. (Contributed by Scott Fenton, 1-Jul-2013.)
 |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ...
 N ) --> RR )
 
Theoremeedimeq 24605 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 24606* 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 24607* 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 24608 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 24609 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 24610* 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 24611* 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 24612* 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 24613 Lemma for colinearalg 24617. 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 24614* Lemma for colinearalg 24617. 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 24615* Lemma for colinearalg 24617. 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 24616* Lemma for colinearalg 24617. 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 24617* An algebraic characterization of colinearity. Note the similarity to brbtwn2 24612. (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 24618* 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 24619* Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 24618. (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 ) )
 
15.4.2.2  Tarski's axioms for geometry for the Euclidean space
 
Theoremaxdimuniq 24620 The unique dimension 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 24621  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 24622 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 24623 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 24624* Lemma for axsegcon 24634. 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 24625* Lemma for axsegcon 24634. 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 24626* Lemma for axsegcon 24634. Show that the square of the distance between two points is nonnegative. (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 24627* Lemma for axsegcon 24634. 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 24628* Lemma for axsegcon 24634. Show that the distance between two points is nonnegative. (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 24629* Lemma for axsegcon 24634. 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 24630* Lemma for axsegcon 24634. 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 24631* Lemma for axsegcon 24634. 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 24632* Lemma for axsegcon 24634. 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 24633* Lemma for axsegcon 24634. Show that the scaling constant from axsegconlem7 24630 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 24634* 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 24635* Lemma for ax5seg 24645. 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 24636* Lemma for ax5seg 24645. 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 24637 Lemma for ax5seg 24645. (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 24638* Lemma for ax5seg 24645. 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 24639* Lemma for ax5seg 24645. 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 24640* Lemma for ax5seg 24645. 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 24641* Lemma for ax5seg 24645. 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 24642 Lemma for ax5seg 24645. 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 24643 Lemma for ax5seg 24645. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 24642. (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 24644* Lemma for ax5seg 24645. Take the calculation in ax5seglem8 24643 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 24645 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 24646 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 24647* Lemma for axpasch 24648. 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 24648* 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 24649 Lemma for axlowdim 24668. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  ( ( 3 ...
 N )  X.  {
 0 } ) : ( 3 ... N )
 --> RR
 
Theoremaxlowdimlem2 24650 Lemma for axlowdim 24668. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  ( ( 1 ... 2 )  i^i  (
 3 ... N ) )  =  (/)
 
Theoremaxlowdimlem3 24651 Lemma for axlowdim 24668. 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 24652 Lemma for axlowdim 24668. 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 24653 Lemma for axlowdim 24668. 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 24654 Lemma for axlowdim 24668. 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 24655 Lemma for axlowdim 24668. 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 24656 Lemma for axlowdim 24668. Calculate 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 24657 Lemma for axlowdim 24668. Calculate 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 24658 Lemma for axlowdim 24668. 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 24659 Lemma for axlowdim 24668. 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 24660 Lemma for axlowdim 24668. Calculate the value of  Q away from its distinguished 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 24661 Lemma for axlowdim 24668. 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 24662 Lemma for axlowdim 24668. Take two possible  Q from axlowdimlem10 24658. They are the same iff their distinguished 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 24663* Lemma for axlowdim 24668. 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 24664* Lemma for axlowdim 24668. 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 24665 Lemma for axlowdim 24668. 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 24666* The lower dimension 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 24667. (Contributed by Scott Fenton, 22-Apr-2013.)
 |-  ( N  e.  NN  ->  E. x  e.  ( EE `  N ) E. y  e.  ( EE `  N ) x  =/=  y )
 
Theoremaxlowdim2 24667* 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 24668* The general lower dimension 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, Alfred Tarski and Steven Givant, Bulletin of 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 24669* Lemma for axeuclid 24670. 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 24670* 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 24671* Lemma for axcont 24683. 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 24672* Lemma for axcont 24683. The idea here is to set up a mapping  F that will allow us to transfer dedekind 9777 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 24673* Lemma for axcont 24683. 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 24674* Lemma for axcont 24683. 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 24675* Lemma for axcont 24683. 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 24676* Lemma for axcont 24683. 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 24677* Lemma for axcont 24683. 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 24678* Lemma for axcont 24683. 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 24679* Lemma for axcont 24683. 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 )
 
Theoremaxcontlem10 24680* Lemma for axcont 24683. Given a handful of assumptions, derive the conclusion of the final theorem. (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 ) )  ->  E. b  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  b  Btwn  <. x ,  y >. )
 
Theoremaxcontlem11 24681* Lemma for axcont 24683. Eliminate the hypotheses from axcontlem10 24680. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  ( ( ( 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 ) )  ->  E. b  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  b  Btwn  <. x ,  y >. )
 
Theoremaxcontlem12 24682* Lemma for axcont 24683. Eliminate the trivial cases from the previous lemmas. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  ( ( ( 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 ) )  ->  E. b  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  b 
 Btwn  <. x ,  y >. )
 
Theoremaxcont 24683* The axiom of continuity. Take two sets of points  A and 
B. If all the points in  A come before the points of  B on a line, then there is a point separating the two. Axiom A11 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  ( ( N  e.  NN  /\  ( A  C_  ( EE `  N ) 
 /\  B  C_  ( EE `  N )  /\  E. a  e.  ( EE
 `  N ) A. x  e.  A  A. y  e.  B  x  Btwn  <. a ,  y >. ) )  ->  E. b  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  b  Btwn  <. x ,  y >. )
 
15.4.2.3  EE^n fulfills Tarski's Axioms
 
Syntaxceeng 24684 Extends class notation with the Tarski geometry structure for  EE ^ N.
 class EEG
 
Definitiondf-eeng 24685* Define the geometry structure for 
EE ^ N. (Contributed by Thierry Arnoux, 24-Aug-2017.)
 |- EEG 
 =  ( n  e. 
 NN  |->  ( { <. (
 Base `  ndx ) ,  ( EE `  n ) >. ,  <. ( dist ` 
 ndx ) ,  ( x  e.  ( EE `  n ) ,  y  e.  ( EE `  n )  |->  sum_ i  e.  (
 1 ... n ) ( ( ( x `  i )  -  (
 y `  i )
 ) ^ 2 ) ) >. }  u.  { <. (Itv `  ndx ) ,  ( x  e.  ( EE `  n ) ,  y  e.  ( EE
 `  n )  |->  { z  e.  ( EE
 `  n )  |  z  Btwn  <. x ,  y >. } ) >. , 
 <. (LineG `  ndx ) ,  ( x  e.  ( EE `  n ) ,  y  e.  ( ( EE `  n ) 
 \  { x }
 )  |->  { z  e.  ( EE `  n )  |  ( z  Btwn  <. x ,  y >.  \/  x  Btwn  <.
 z ,  y >.  \/  y  Btwn  <. x ,  z >. ) } ) >. } ) )
 
Theoremeengv 24686* The value of the Euclidean geometry for dimension  N. (Contributed by Thierry Arnoux, 15-Mar-2019.)
 |-  ( N  e.  NN  ->  (EEG `  N )  =  ( { <. ( Base ` 
 ndx ) ,  ( EE `  N ) >. , 
 <. ( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE
 `  N )  |->  sum_ i  e.  ( 1 ...
 N ) ( ( ( x `  i
 )  -  ( y `
  i ) ) ^ 2 ) )
 >. }  u.  { <. (Itv `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N )  |->  { z  e.  ( EE `  N )  |  z  Btwn  <. x ,  y >. } ) >. , 
 <. (LineG `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( ( EE `  N ) 
 \  { x }
 )  |->  { z  e.  ( EE `  N )  |  ( z  Btwn  <. x ,  y >.  \/  x  Btwn  <.
 z ,  y >.  \/  y  Btwn  <. x ,  z >. ) } ) >. } ) )
 
Theoremeengstr 24687 The Euclidean geometry as a structure. (Contributed by Thierry Arnoux, 15-Mar-2019.)
 |-  ( N  e.  NN  ->  (EEG `  N ) Struct  <.
 1 , ; 1 7 >. )
 
Theoremeengbas 24688 The Base of the Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)
 |-  ( N  e.  NN  ->  ( EE `  N )  =  ( Base `  (EEG `  N )
 ) )
 
Theoremebtwntg 24689 The betweenness relation used in the Tarski structure for the Euclidean geometry is the same as 
Btwn. (Contributed by Thierry Arnoux, 15-Mar-2019.)
 |-  ( ph  ->  N  e.  NN )   &    |-  P  =  (
 Base `  (EEG `  N ) )   &    |-  I  =  (Itv `  (EEG `  N )
 )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   =>    |-  ( ph  ->  ( Z  Btwn  <. X ,  Y >.  <->  Z  e.  ( X I Y ) ) )
 
Theoremecgrtg 24690 The congruence relation used in the Tarski structure for the Euclidean geometry is the same as Cgr. (Contributed by Thierry Arnoux, 15-Mar-2019.)
 |-  ( ph  ->  N  e.  NN )   &    |-  P  =  (
 Base `  (EEG `  N ) )   &    |-  .-  =  ( dist `  (EEG `  N ) )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   =>    |-  ( ph  ->  (
 <. A ,  B >.Cgr <. C ,  D >.  <->  ( A  .-  B )  =  ( C  .-  D ) ) )
 
Theoremelntg 24691* The line definition in the Tarski structure for the Euclidean geometry. (Contributed by Thierry Arnoux, 7-Apr-2019.)
 |-  P  =  ( Base `  (EEG `  N )
 )   &    |-  I  =  (Itv `  (EEG `  N ) )   =>    |-  ( N  e.  NN  ->  (LineG `  (EEG `  N ) )  =  ( x  e.  P ,  y  e.  ( P  \  { x } )  |->  { z  e.  P  |  ( z  e.  ( x I y )  \/  x  e.  ( z I y )  \/  y  e.  ( x I z ) ) } ) )
 
Theoremeengtrkg 24692 The geometry structure for  EE ^ N is a Tarski geometry (Contributed by Thierry Arnoux, 15-Mar-2019.)
 |-  ( N  e.  NN  ->  (EEG `  N )  e. TarskiG )
 
Theoremeengtrkge 24693 The geometry structure for  EE ^ N is a Euclidean geometry (Contributed by Thierry Arnoux, 15-Mar-2019.)
 |-  ( N  e.  NN  ->  (EEG `  N )  e. TarskiGE )
 
PART 16  GRAPH THEORY



To give an overview of the definitions and terms used in the context of graph theory, a glossary is provided in the following, mainly according to definitions in [Bollobas] p. 1-8 or in [Diestel] p. 2-28. Although this glossary concentrates on undirected graphs, many of the concepts are also useful for directed graphs.

Basic kinds of graphs:

TermReferenceDefinitionRemarks
Undirected hypergraph df-uhgra 24696 an ordered pair  <. V ,  E >. of a set  V and a function  E into the powerset of  V ( ran  E  C_  ( ~P V )).
An element of  V is called "vertex", an element of  ran  E is called "edge", the function  E is called the "edge-function" .
In this most general definition of a graph, an "edge" may connect three or more vertices with each other, see [Berge] p. 1.
In Wikipedia "Hypergraph", see https://en.wikipedia.org/wiki/Hypergraph (18-Jan-2020) such a hypergraph is called "non-simple hypergraph", "multiple hypergraph" or "multi-hypergraphs". According to Wikipedia "Incidence structure", see https://en.wikipedia.org/wiki/Incidence_structure (18-Jan-2020) "Each hypergraph [...] can be regarded as an incidence structure in which the [vertices] play the role of "points", the corresponding family of [edges] plays the role of "lines" and the incidence relation is set membership".

If a graph is represented by a class variable, e.g.  G, the edges of this graph are often represented by the function value  ( Edges  `  G ). If the graph is given as pair  <. V ,  E >., however,  ( Edges  `  <. V ,  E >. ) or preferably  ( V Edges  E ) is only used to talk about edges more explicitly. Otherwise,  ran  E is used, because this is much shorter.
Notice that by using  ( Edges  `  G ) the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. Therefore, this representation will only be used for undirected simple graphs.
For the set of vertices, a function "Vertices" could have been defined analogously. But "( Vertices ` G )" would have been exactly the same as  ( 1st `  G ), so the latter is used to denote the set of vertices if the graph is represented by a class variable.
Undirected simple hypergraph df-ushgra 24697 an ordered pair  <. V ,  E >. of a set  V and a one-to-one function  E into the powerset of  V ( ran  E  C_  ( ~P V )). See also Wikipedia "Hypergraph", https://en.wikipedia.org/wiki/Hypergraph (18-Jan-2020). This is how a "hypergraph" is defined in Section I.1 in [Bollobas] p. 7 or the definition in section 1.10 in [Diestel] p. 27. A simple hypergraph has at most one edge between the same vertices, hence a multigraph needs not to be a hypergraph.
According to [Berge] p. 1, "A simple hypergraph (or "Sperner family") is a hypergraph H = { E1, E2, ..., Em } such that Ei C_ Ej => i = j". By this definition, a simple hypergraph cannot contain the edges E1 = { v1 , v2 } and E2 = { v1, v2, v3 }, because E1 C_ E2, but 1 =/= 2.
Undirected multigraph df-umgra 24717 a graph  <. V ,  E >. such that  E is a function into the set of (proper or not proper) unordered pairs of  V.A proper unordered pair contains two different elements, a not proper unordered pair contains two times the same element, so it is a singleton (see preqsn 4154).
According to the definition in Section I.1 in [Bollobas] p. 7, "In a multigraph both multiple edges [joining two vertices] and multiple loops [joining a vertex to itself] are allowed", or according to [Diestel] p. 28, "A multigraph is a pair (V,E) of disjoint sets (of vertices and edges) together with a map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end.".
Undirected simple graph with loops df-uslgra 24736 a graph  <. V ,  E >. such that  E is a one-to-one function into the set of (proper or not proper) unordered pairs of  V.This means that there is at most one edge between two vertices, and at most one loop from a vertex to itself.
Undirected simple graph without loops (in short "simple graph") df-usgra 24737 a graph  <. V ,  E >. such that  E is a one-to-one function into the set of (proper) unordered pairs of  V.An ordered pair  <. V ,  E >. of two distinct sets  V and  E (the "usual" definition of a "graph", see, for example, the definition in section I.1 of [Bollobas] p. 1 or in section 1.1 of [Diestel] p. 2) can be identified with an undirected simple graph without loops by "indexing" the edges with themselves, see ausisusgra 24759.
Finite graph---a graph  <. V ,  E >. with finite sets  V and  E.See definitions in [Bollobas] p. 1 or [Diestel] p. 2.
In simple graphs,  E is finite if  V is finite, see usgrafis 24819. The number of edges is limited by  ( n  _C  2 ) (or " n choose 2") with  n  =  ( # `  V ), see usgramaxsize 24891. Analogously, the number of edges of an undirected simple graph with loops is limited by  ( ( n  +  1 )  _C  2 ). In multigraphs, however,  E can be infinite although  V is finite.
Graph of finite size---a graph  <. V ,  E >. with finite set  E, i.e. with a finite number of edges.A graph can be of finite size although  V is infinite.


Terms and properties of graphs:
TermReferenceDefinitionRemarks
Edge joining (two) vertices --- An edge  e  e.  ran  E "joins" the vertices v1, v2, ... vn ( n  e.  NN) if  e = { v1, v2, ... vn }. If  n  =  1,  e = { v1 } is a "loop", if  n  =  2,  e = { v1 , v2 } is an edge as it is usually defined, see definition in Section I.1 in [Bollobas] p. 1.
(Two) Endvertices of an edge see definition in Section I.1 in [Bollobas] p. 1. If an edge  e  e.  ran  E joins the vertices v1, v2, ... vn ( n  e.  NN), then the vertices v1, v2, ... vn are called the "endvertices" of the edge  e.
(Two) Adjacent vertices see definition in Section I.1 in [Bollobas] p. 1/2. The vertices v1, v2, ... vn ( n  e.  NN) are "adjacent" if there is an edge e = { v1, v2, ... vn } joining these vertices. In this case, the vertices are "incident" with the edge e (see definition in Section I.1 in [Bollobas] p. 2) or "connected" by the edge e.
Edge ending at a vertex An edge  e  e.  ran  E is "ending" at a vertex  v if the vertex is an endvertex of the edge:  v  e.  e. In other words, the vertex  v is incident with the edge  e.
(Two) Adjacent edges The edges e0, e1, ... en ( n  e.  NN) are "adjacent" if they have exactly one common endvertex. Generalization of definition in Section I.1 in [Bollobas] p. 2.
Order of a graph see definition in Section I.1 in [Bollobas] p. 3 the "order" of a graph  <. V ,  E >. is the number of vertices in the graph ( ( # `  V )).
Size of a graph see definition in Section I.1 in [Bollobas] p. 3 the "size" of a graph  <. V ,  E >. is the number of edges in the graph ( ( # `  E )). Or, for simple graphs  G:  ( # `  ( Edges  `  G ) )).
Neighborhood of a vertex df-nbgra 24824 resp. definition in Section I.1 in [Bollobas] p. 3 A vertex connected with a vertex  v by an edge is called a "neighbor" of the vertex  v. The set of neighbors of a vertex  v is called the "neighborhood" (or "open neighborhood") of the vertex  v. The "closed neighborhood" is the union of the (open) neighborhood of the vertex  v with  { v }.
Degree of a vertex df-vdgr 25298 The "degree" of a vertex is the number of the edges ending at this vertex. In a simple graph, the degree of a vertex is the number of neighbors of this vertex, see definition in Section I.1 in [Bollobas] p. 3
Isolated vertex usgravd0nedg 25322 A vertex is called "isolated" if it is not an endvertex of any edge, thus having degree 0.
Universal vertex df-uvtx 24826 A vertex is called "universal" if it is connected with every other vertex of the graph by an edge, thus having degree  ( # `  V ).


Special kinds of graphs:
TermReferenceDefinitionRemarks
Complete graph df-cusgra 24825 A graph is called "complete" if each pair of vertices is connected by an edge. The size of a complete undirected simple graph of order  n is  ( n  _C  2 ) (or " n choose 2"), see cusgrasize 24882.
Empty graph umgra0 24729 and usgra0 24774 A graph is called "empty" if it has no edges.
Null graph usgra0v 24775 A graph is called the "null graph" if it has no vertices (and therefore also no edges).
Trivial graph usgra1v 24794 A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph df-conngra 25074 resp. definition in Section I.1 in [Bollobas] p. 6 A graph is called "connected" if for each pair of vertices there is a path between these vertices.


For the terms "Path", "Walk", "Trail", "Circuit", "Cycle" see the remarks below and the definitions in Section I.1 in [Bollobas] p. 4-5.
 
16.1  Undirected graphs - basics
 
16.1.1  Undirected hypergraphs
 
Syntaxcuhg 24694 Extend class notation with undirected hypergraphs.
 class UHGrph
 
Syntaxcushg 24695 Extend class notation with undirected simple hypergraphs.
 class USHGrph
 
Definitiondf-uhgra 24696* Define the class of all undirected hypergraphs. An undirected hypergraph is a pair of a set and a function into the powerset of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |- UHGrph  =  { <. v ,  e >.  |  e : dom  e
 --> ( ~P v  \  { (/) } ) }
 
Definitiondf-ushgra 24697* Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a pair of a set and an injective (one-to-one) function into the powerset of this set (the empty set excluded). (Contributed by AV, 19-Jan-2020.)
 |- USHGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> ( ~P v  \  { (/) } ) }
 
Theoremreluhgra 24698 The class of all undirected hypergraphs is a relation. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |- 
 Rel UHGrph
 
Theoremrelushgra 24699 The class of all undirected simple hypergraphs is a relation. (Contributed by AV, 19-Jan-2020.)
 |- 
 Rel USHGrph
 
Theoremuhgrav 24700 The classes of vertices and edges of an undirected hypergraph are sets. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( V UHGrph  E  ->  ( V  e.  _V  /\  E  e.  _V )
 )
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