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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxsegcon 23901* 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 23902* Lemma for ax5seg 23912. 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 23903* Lemma for ax5seg 23912. 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 23904 Lemma for ax5seg 23912. (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 23905* Lemma for ax5seg 23912. 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 23906* Lemma for ax5seg 23912. 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 23907* Lemma for ax5seg 23912. 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 23908* Lemma for ax5seg 23912. 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 23909 Lemma for ax5seg 23912. 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 23910 Lemma for ax5seg 23912. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 23909. (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 23911* Lemma for ax5seg 23912. Take the calculation in ax5seglem8 23910 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 23912 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 23913 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 23914* Lemma for axpasch 23915. 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 23915* 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 23916 Lemma for axlowdim 23935. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  ( ( 3 ...
 N )  X.  {
 0 } ) : ( 3 ... N )
 --> RR
 
Theoremaxlowdimlem2 23917 Lemma for axlowdim 23935. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  ( ( 1 ... 2 )  i^i  (
 3 ... N ) )  =  (/)
 
Theoremaxlowdimlem3 23918 Lemma for axlowdim 23935. 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 23919 Lemma for axlowdim 23935. 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 23920 Lemma for axlowdim 23935. 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 23921 Lemma for axlowdim 23935. 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 23922 Lemma for axlowdim 23935. 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 23923 Lemma for axlowdim 23935. 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 23924 Lemma for axlowdim 23935. 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 23925 Lemma for axlowdim 23935. 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 23926 Lemma for axlowdim 23935. 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 23927 Lemma for axlowdim 23935. 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 23928 Lemma for axlowdim 23935. 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 23929 Lemma for axlowdim 23935. Take two possible  Q from axlowdimlem10 23925. 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 23930* Lemma for axlowdim 23935. 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 23931* Lemma for axlowdim 23935. 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 23932 Lemma for axlowdim 23935. 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 23933* 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 23934. (Contributed by Scott Fenton, 22-Apr-2013.)
 |-  ( N  e.  NN  ->  E. x  e.  ( EE `  N ) E. y  e.  ( EE `  N ) x  =/=  y )
 
Theoremaxlowdim2 23934* 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 23935* 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 23936* Lemma for axeuclid 23937. 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 23937* 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 23938* Lemma for axcont 23950. 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 23939* Lemma for axcont 23950. The idea here is to set up a mapping  F that will allow us to transfer dedekind 9734 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 23940* Lemma for axcont 23950. 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 23941* Lemma for axcont 23950. 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 23942* Lemma for axcont 23950. 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 23943* Lemma for axcont 23950. 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 23944* Lemma for axcont 23950. 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 23945* Lemma for axcont 23950. 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 23946* Lemma for axcont 23950. 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 23947* Lemma for axcont 23950. 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 23948* Lemma for axcont 23950. Eliminate the hypotheses from axcontlem10 23947. (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 23949* Lemma for axcont 23950. 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 23950* 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 23951 Extends class notation with the Tarski geometry structure for  EE ^ N.
 class EEG
 
Definitiondf-eeng 23952* 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 23953* 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 23954 The Euclidean geometry as a structure. (Contributed by Thierry Arnoux, 15-Mar-2019.)
 |-  ( N  e.  NN  ->  (EEG `  N ) Struct  <.
 1 , ; 1 7 >. )
 
Theoremeengbas 23955 The Base of the Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)
 |-  ( N  e.  NN  ->  ( EE `  N )  =  ( Base `  (EEG `  N )
 ) )
 
Theoremebtwntg 23956 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 23957 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 23958* 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 23959 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 23960 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. 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 23962 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, compare with the definition in Section I.1 in [Bollobas] p. 7.
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 multigraph df-umgra 23978 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 4204).
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".
Undirected simple graph with loops df-uslgra 23997 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 23998 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 in [Bollobas] p. 1) can be identified with an undirected simple graph without loops by "indexing" the edges with themselves, see ausisusgra 24020.
Finite graph---a graph  <. V ,  E >. with finite sets  V and  E.In simple graphs,  E is finite if  V is finite, see usgrafis 24079. The number of edges is limited by  ( n  _C  2 ) (or " n choose 2") with  n  =  ( # `  V ), see usgramaxsize 24151. 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 24084 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 24558 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 24582 A vertex is called "isolated" if it is not an endvertex of any edge, thus having degree 0.
Universal vertex df-uvtx 24086 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 24085 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 24142.
Empty graph umgra0 23990 and usgra0 24034 A graph is called "empty" if it has no edges.
Null graph usgra0v 24035 A graph is called the "null graph" if it has no vertices (and therefore also no edges).
Trivial graph usgra1v 24054 A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph df-conngra 24334 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 23961 Extend class notation with undirected hypergraphs.
 class UHGrph
 
Definitiondf-uhgra 23962* 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  \  { (/) } ) }
 
Theoremreluhgra 23963 The class of all undirected hypergraphs is a relation. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |- 
 Rel UHGrph
 
Theoremuhgrav 23964 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 )
 )
 
Theoremuhgraopelvv 23965 An undirected hypergraph is a member in the universal class of ordered pairs. (Contributed by AV, 3-Jan-2020.)
 |-  ( G  e. UHGrph  ->  G  e.  ( _V  X.  _V ) )
 
Theoremisuhgra 23966 The property of being an undirected hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V UHGrph  E  <->  E : dom  E --> ( ~P V  \  { (/) } )
 ) )
 
Theoremuhgraf 23967 The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( V UHGrph  E  ->  E : dom  E --> ( ~P V  \  { (/) } )
 )
 
Theoremuhgrafun 23968 The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( V UHGrph  E  ->  Fun 
 E )
 
Theoremuhgraop 23969 The property of being an undirected hypergraph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see [Bollobas] p. 1. (Contributed by AV, 1-Jan-2020.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( <. V ,  E >.  e. UHGrph  <->  E : dom  E --> ( ~P V  \  { (/)
 } ) ) )
 
Theoremuhgracl 23970 The property of being an undirected hypergraph represented by a class. This representation is useful if the set of vertices and the edge function is/needs not to be known. (Contributed by AV, 1-Jan-2020.)
 |-  ( G  e. UHGrph  ->  ( 2nd `  G ) : dom  ( 2nd `  G )
 --> ( ~P ( 1st `  G )  \  { (/)
 } ) )
 
Theoremuhgrass 23971 An edge is a subset of vertices, analogous to umgrass 23984. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( ( V UHGrph  E  /\  F  e.  dom  E )  ->  ( E `  F )  C_  V )
 
Theoremuhgraeq12d 23972 Equality of hypergraphs. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( V  =  W  /\  E  =  F )
 )  ->  ( V UHGrph  E  <->  W UHGrph  F ) )
 
Theoremuhgrares 23973 A subgraph of a hypergraph (formed by removing some edges from the original graph) is a hypergraph, analogous to umgrares 23989. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( V UHGrph  E  ->  V UHGrph 
 ( E  |`  A ) )
 
Theoremuhgra0 23974 The empty graph, with vertices but no edges, is a hypergraph, analogous to umgra0 23990. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( V  e.  W  ->  V UHGrph  (/) )
 
Theoremuhgra0v 23975 The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( (/) UHGrph  E  <->  E  =  (/) )
 
Theoremuhgraun 23976 The union of two (undirected) hypergraphs (with the same vertex set): If  <. V ,  E >. and 
<. V ,  F >. are hypergraphs, then  <. V ,  E  u.  F >. is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices), analogous to umgraun 23993. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UHGrph  E )   &    |-  ( ph  ->  V UHGrph  F )   =>    |-  ( ph  ->  V UHGrph 
 ( E  u.  F ) )
 
16.1.2  Undirected multigraphs
 
Syntaxcumg 23977 Extend class notation with undirected multigraphs.
 class UMGrph
 
Definitiondf-umgra 23978* Define the class of all undirected multigraphs. A multigraph is a pair  <. V ,  E >. where  E is a function into subsets of  V of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- UMGrph  =  { <. v ,  e >.  |  e : dom  e
 --> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
 
Theoremrelumgra 23979 The class of all undirected multigraphs is a relation. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- 
 Rel UMGrph
 
Theoremisumgra 23980* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremwrdumgra 23981* The property of being an undirected multigraph, expressing the edges as "words". (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V  e.  W  /\  E  e. Word  X )  ->  ( V UMGrph  E  <->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremumgraf2 23982* The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgraf 23983 without explicitly specified domain of the edge function (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremumgraf 23983* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A ) 
 ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremumgrass 23984 An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  C_  V )
 
Theoremumgran0 23985 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  =/=  (/) )
 
Theoremumgrale 23986 An edge has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( # `  ( E `  F ) ) 
 <_  2 )
 
Theoremumgrafi 23987 An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  e.  Fin )
 
Theoremumgraex 23988* An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  E. x  e.  V  E. y  e.  V  ( E `  F )  =  { x ,  y } )
 
Theoremumgrares 23989 A subgraph of a graph (formed by removing some edges from the original graph) is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V UMGrph  E  ->  V UMGrph 
 ( E  |`  A ) )
 
Theoremumgra0 23990 The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V  e.  W  ->  V UMGrph  (/) )
 
Theoremumgra1 23991 The graph with one edge. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V )
 )  ->  V UMGrph  { <. A ,  { B ,  C } >. } )
 
Theoremumisuhgra 23992 An undirected multigraph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
 |-  ( V UMGrph  E  ->  V UHGrph  E )
 
Theoremumgraun 23993 The union of two (undirected) multigraphs (with the same vertex set): If  <. V ,  E >. and 
<. V ,  F >. are graphs, then  <. V ,  E  u.  F >. is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UMGrph  E )   &    |-  ( ph  ->  V UMGrph  F )   =>    |-  ( ph  ->  V UMGrph 
 ( E  u.  F ) )
 
16.1.3  Undirected simple graphs
 
16.1.3.1  Undirected simple graphs - basics
 
Syntaxcuslg 23994 Extend class notation with undirected (simple) graphs with loops.
 class USLGrph
 
Syntaxcusg 23995 Extend class notation with undirected (simple) graphs (without loops).
 class USGrph
 
Syntaxcedg 23996 Extend class notation with the set of edges (of an undirected simple graph).
 class Edges
 
Definitiondf-uslgra 23997* Define the class of all undirected simple graphs with loops. An undirected simple graph with loops is a special undirected multigraph  <. V ,  E >. where  E is an injective (one-to-one) function into subsets of  V of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a multigraph, there is at most one edge between two vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- USLGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
 
Definitiondf-usgra 23998* Define the class of all undirected simple graphs without loops. An undirected simple graph without loops is a special undirected simple graph  <. V ,  E >. where 
E is an injective (one-to-one) function into subsets of  V of cardinality two, representing the two vertices incident to the edge. Such graphs are usually simply called "undirected graphs", so if only the term "undirected graph" is used, an undirected simple graph without loops is meant. Therefore, an undirected graph has no loops (edges a vertex to itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- USGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
 
Theoremreluslgra 23999 The class of all undirected simple graph with loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- 
 Rel USLGrph
 
Theoremrelusgra 24000 The class of all undirected simple graph without loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- 
 Rel USGrph
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