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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremax5seglem5 23901* Lemma for ax5seg 23906. 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 23902* Lemma for ax5seg 23906. 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 23903 Lemma for ax5seg 23906. 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 23904 Lemma for ax5seg 23906. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 23903. (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 23905* Lemma for ax5seg 23906. Take the calculation in ax5seglem8 23904 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 23906 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 23907 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 23908* Lemma for axpasch 23909. 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 23909* 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 23910 Lemma for axlowdim 23929. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  (
 ( 3 ... N )  X.  { 0 } ) : ( 3
 ... N ) --> RR
 
Theoremaxlowdimlem2 23911 Lemma for axlowdim 23929. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  (
 ( 1 ... 2
 )  i^i  ( 3 ... N ) )  =  (/)
 
Theoremaxlowdimlem3 23912 Lemma for axlowdim 23929. 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 23913 Lemma for axlowdim 23929. 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 23914 Lemma for axlowdim 23929. 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 23915 Lemma for axlowdim 23929. 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 23916 Lemma for axlowdim 23929. 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 23917 Lemma for axlowdim 23929. Calulate the value of  P at three. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   =>    |-  ( P `  3
 )  =  -u 1
 
Theoremaxlowdimlem9 23918 Lemma for axlowdim 23929. Calulate the value of  P away from three. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   =>    |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  3
 )  ->  ( P `  K )  =  0 )
 
Theoremaxlowdimlem10 23919 Lemma for axlowdim 23929. 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 23920 Lemma for axlowdim 23929. 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 23921 Lemma for axlowdim 23929. Calculate the value of  Q away from its distunguished point. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  (
 ( ( 1 ...
 N )  \  {
 ( I  +  1 ) } )  X.  { 0 } ) )   =>    |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  ( I  +  1 )
 )  ->  ( Q `  K )  =  0 )
 
Theoremaxlowdimlem13 23922 Lemma for axlowdim 23929. 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 23923 Lemma for axlowdim 23929. Take two possible  Q from axlowdimlem10 23919. They are the same iff their distunguished values are the same. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  (
 ( ( 1 ...
 N )  \  {
 ( I  +  1 ) } )  X.  { 0 } ) )   &    |-  R  =  ( { <. ( J  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N ) 
 \  { ( J  +  1 ) }
 )  X.  { 0 } ) )   =>    |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  -  1 ) ) 
 /\  J  e.  (
 1 ... ( N  -  1 ) ) ) 
 ->  ( Q  =  R  ->  I  =  J ) )
 
Theoremaxlowdimlem15 23924* Lemma for axlowdim 23929. 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 23925* Lemma for axlowdim 23929. 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 23926 Lemma for axlowdim 23929. 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 23927* The lower dimensional axiom for one dimension. In any dimension, there are at least two distinct points. Theorem 3.13 of [Schwabhauser] p. 32, where it is derived from axlowdim2 23928. (Contributed by Scott Fenton, 22-Apr-2013.)
 |-  ( N  e.  NN  ->  E. x  e.  ( EE
 `  N ) E. y  e.  ( EE `  N ) x  =/=  y )
 
Theoremaxlowdim2 23928* 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 23929* The general lower dimensional axiom. Take a dimension  N greater than or equal to three. Then, there are three non-colinear points in  N dimensional space that are equidistant from  N  -  1 distinct points. Derived from remarks in "Tarski's System of Geometry", by Alfred Tarski and Steven Givant, Bull. Symbolic Logic Volume 5, Number 2 (1999), 175-214. (Contributed by Scott Fenton, 22-Apr-2013.)
 |-  ( N  e.  ( ZZ>= `  3 )  ->  E. p E. x  e.  ( EE `  N ) E. y  e.  ( EE `  N ) E. z  e.  ( EE `  N ) ( p :
 ( 1 ... ( N  -  1 ) )
 -1-1-> ( EE `  N )  /\  A. i  e.  ( 2 ... ( N  -  1 ) ) ( <. ( p `  1 ) ,  x >.Cgr
 <. ( p `  i
 ) ,  x >.  /\ 
 <. ( p `  1
 ) ,  y >.Cgr <.
 ( p `  i
 ) ,  y >.  /\ 
 <. ( p `  1
 ) ,  z >.Cgr <.
 ( p `  i
 ) ,  z >. ) 
 /\  -.  ( x  Btwn  <. y ,  z >.  \/  y  Btwn  <. z ,  x >.  \/  z  Btwn  <. x ,  y >. ) ) )
 
Theoremaxeuclidlem 23930* Lemma for axeuclid 23931. 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 23931* 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 23932* Lemma for axcont 23944. 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 23933* Lemma for axcont 23944. The idea here is to set up a mapping  F that will allow us to transfer dedekind 23418 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 23934* Lemma for axcont 23944. 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 23935* Lemma for axcont 23944. 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 23936* Lemma for axcont 23944. 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 23937* Lemma for axcont 23944. 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 23938* Lemma for axcont 23944. 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 23939* Lemma for axcont 23944. 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 23940* Lemma for axcont 23944. 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 23941* Lemma for axcont 23944. 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 23942* Lemma for axcont 23944. Eliminate the hypotheses from axcontlem10 23941. (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 23943* Lemma for axcont 23944. 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 23944* 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 >. )
 
16.7.32  Congruence properties
 
Syntaxcofs 23945 Declare the syntax for the outer five segment configuration.
 class  OuterFiveSeg
 
Definitiondf-ofs 23946* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 23906). See brofs 23968 and 5segofs 23969 for how it is used. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.)
 |-  OuterFiveSeg  =  { <. p ,  q >.  |  E. n  e.  NN  E. a  e.  ( EE `  n ) E. b  e.  ( EE `  n ) E. c  e.  ( EE `  n ) E. d  e.  ( EE `  n ) E. x  e.  ( EE `  n ) E. y  e.  ( EE `  n ) E. z  e.  ( EE `  n ) E. w  e.  ( EE `  n ) ( p  =  <. <. a ,  b >. ,  <. c ,  d >. >.  /\  q  =  <.
 <. x ,  y >. , 
 <. z ,  w >. >.  /\  ( ( b  Btwn  <.
 a ,  c >.  /\  y  Btwn  <. x ,  z >. )  /\  ( <. a ,  b >.Cgr <. x ,  y >.  /\ 
 <. b ,  c >.Cgr <.
 y ,  z >. ) 
 /\  ( <. a ,  d >.Cgr <. x ,  w >.  /\  <. b ,  d >.Cgr
 <. y ,  w >. ) ) ) }
 
Theoremcgrrflx2d 23947 Deduction form of axcgrrflx 23882. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   =>    |-  ( ph  ->  <. A ,  B >.Cgr <. B ,  A >. )
 
Theoremcgrtr4d 23948 Deduction form of axcgrtr 23883. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ph  ->  <. A ,  B >.Cgr <. C ,  D >. )   &    |-  ( ph  ->  <. A ,  B >.Cgr <. E ,  F >. )   =>    |-  ( ph  ->  <. C ,  D >.Cgr <. E ,  F >. )
 
Theoremcgrtr4and 23949 Deduction form of axcgrtr 23883. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. C ,  D >. )   &    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. E ,  F >. )   =>    |-  ( ( ph  /\  ps )  ->  <. C ,  D >.Cgr
 <. E ,  F >. )
 
Theoremcgrrflx 23950 Reflexivity law for congruence. Theorem 2.1 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.Cgr <. A ,  B >. )
 
Theoremcgrrflxd 23951 Deduction form of cgrrflx 23950. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   =>    |-  ( ph  ->  <. A ,  B >.Cgr <. A ,  B >. )
 
Theoremcgrcomim 23952 Congruence commutes on the two sides. Implication version. Theorem 2.2 of [Schwabhauser] p. 27. (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 ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  ->  <. C ,  D >.Cgr <. A ,  B >. ) )
 
Theoremcgrcom 23953 Congruence commutes between the two sides. (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 ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  <->  <. C ,  D >.Cgr <. A ,  B >. ) )
 
Theoremcgrcomand 23954 Deduction form of cgrcom 23953. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. C ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  <. C ,  D >.Cgr
 <. A ,  B >. )
 
Theoremcgrtr 23955 Transitivity law for congruence. Theorem 2.3 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 24-Sep-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 >.  /\ 
 <. C ,  D >.Cgr <. E ,  F >. ) 
 ->  <. A ,  B >.Cgr
 <. E ,  F >. ) )
 
Theoremcgrtrand 23956 Deduction form of cgrtr 23955. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. C ,  D >. )   &    |-  ( ( ph  /\  ps )  ->  <. C ,  D >.Cgr
 <. E ,  F >. )   =>    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. E ,  F >. )
 
Theoremcgrtr3 23957 Transitivity law for congruence. (Contributed by Scott Fenton, 7-Oct-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 <. E ,  F >.  /\ 
 <. C ,  D >.Cgr <. E ,  F >. ) 
 ->  <. A ,  B >.Cgr
 <. C ,  D >. ) )
 
Theoremcgrtr3and 23958 Deduction form of cgrtr3 23957. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. E ,  F >. )   &    |-  ( ( ph  /\  ps )  ->  <. C ,  D >.Cgr
 <. E ,  F >. )   =>    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. C ,  D >. )
 
Theoremcgrcoml 23959 Congruence commutes on the left. Biconditional version of Theorem 2.4 of [Schwabhauser] p. 27. (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 ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  <->  <. B ,  A >.Cgr <. C ,  D >. ) )
 
Theoremcgrcomr 23960 Congruence commutes on the right. Biconditional version of Theorem 2.5 of [Schwabhauser] p. 27. (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 ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  <->  <. A ,  B >.Cgr <. D ,  C >. ) )
 
Theoremcgrcomlr 23961 Congruence commutes on both sides. (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 ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  <->  <. B ,  A >.Cgr <. D ,  C >. ) )
 
Theoremcgrcomland 23962 Deduction form of cgrcoml 23959. (Contributed by Scott Fenton, 14-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. C ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  <. B ,  A >.Cgr
 <. C ,  D >. )
 
Theoremcgrcomrand 23963 Deduction form of cgrcoml 23959. (Contributed by Scott Fenton, 14-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. C ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. D ,  C >. )
 
Theoremcgrcomlrand 23964 Deduction form of cgrcomlr 23961. (Contributed by Scott Fenton, 14-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. C ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  <. B ,  A >.Cgr
 <. D ,  C >. )
 
Theoremcgrtriv 23965 Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  A >.Cgr <. B ,  B >. )
 
Theoremcgrid2 23966 Identity law for congruence. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( <. A ,  A >.Cgr
 <. B ,  C >.  ->  B  =  C )
 )
 
Theoremcgrdegen 23967 Two congruent segments are either both degenrate or both non-degenerate. (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 ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  ->  ( A  =  B  <->  C  =  D ) ) )
 
Theorembrofs 23968 Binary relationship form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-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 >. , 
 <. C ,  D >. >.  OuterFiveSeg  <. <. E ,  F >. , 
 <. G ,  H >. >.  <->  ( ( 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 >. ) ) ) )
 
Theorem5segofs 23969 Rephrase ax5seg 23906 using the outer five segment predicate. Theorem 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-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 >. , 
 <. C ,  D >. >.  OuterFiveSeg  <. <. E ,  F >. , 
 <. G ,  H >. >.  /\  A  =/=  B ) 
 ->  <. C ,  D >.Cgr
 <. G ,  H >. ) )
 
Theoremofscom 23970 The outer five segment predicate commutes. (Contributed by Scott Fenton, 26-Sep-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 >. , 
 <. C ,  D >. >.  OuterFiveSeg  <. <. E ,  F >. , 
 <. G ,  H >. >.  <->  <. <. E ,  F >. , 
 <. G ,  H >. >.  OuterFiveSeg  <. <. A ,  B >. , 
 <. C ,  D >. >.
 ) )
 
Theoremcgrextend 23971 Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (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 ) ) )  ->  ( (
 ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. ) 
 /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
 <. E ,  F >. ) )  ->  <. A ,  C >.Cgr <. D ,  F >. ) )
 
Theoremcgrextendand 23972 Deduction form of cgrextend 23971. (Contributed by Scott Fenton, 14-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  B  Btwn  <. A ,  C >. )   &    |-  ( ( ph  /\ 
 ps )  ->  E  Btwn  <. D ,  F >. )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. D ,  E >. )   &    |-  ( ( ph  /\ 
 ps )  ->  <. B ,  C >.Cgr <. E ,  F >. )   =>    |-  ( ( ph  /\  ps )  ->  <. A ,  C >.Cgr
 <. D ,  F >. )
 
Theoremsegconeq 23973 Two points that satsify the conclusion of axsegcon 23895 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N ) 
 /\  Y  e.  ( EE `  N ) ) )  ->  ( ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
 <. B ,  C >. ) 
 /\  ( A  Btwn  <. Q ,  Y >.  /\ 
 <. A ,  Y >.Cgr <. B ,  C >. ) )  ->  X  =  Y ) )
 
Theoremsegconeu 23974* Existential uniqueness version of segconeq 23973. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) 
 /\  C  =/=  D ) )  ->  E! r  e.  ( EE `  N ) ( D  Btwn  <. C ,  r >.  /\ 
 <. D ,  r >.Cgr <. A ,  B >. ) )
 
16.7.33  Betweenness properties
 
Theorembtwntriv2 23975 Betweeness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  B  Btwn  <. A ,  B >. )
 
Theorembtwncomim 23976 Betweeness commutes. Implication version. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Btwn  <. B ,  C >.  ->  A  Btwn  <. C ,  B >. ) )
 
Theorembtwncom 23977 Betweeness commutes. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Btwn  <. B ,  C >. 
 <->  A  Btwn  <. C ,  B >. ) )
 
Theorembtwncomand 23978 Deduction form of btwncom 23977. (Contributed by Scott Fenton, 14-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  A  Btwn  <. B ,  C >. )   =>    |-  ( ( ph  /\  ps )  ->  A  Btwn  <. C ,  B >. )
 
Theorembtwntriv1 23979 Betweeness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  A  Btwn  <. A ,  B >. )
 
Theorembtwnswapid 23980 If you can swap the first two arguments of a betweenness statement, then those arguments are identical. Theorem 3.4 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( ( A  Btwn  <. B ,  C >.  /\  B  Btwn  <. A ,  C >. )  ->  A  =  B ) )
 
Theorembtwnswapid2 23981 If you can swap arguments one and three of a betweenness statement, then those arguments are identical. (Contributed by Scott Fenton, 7-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( ( A  Btwn  <. B ,  C >.  /\  C  Btwn  <. B ,  A >. )  ->  A  =  C ) )
 
Theorembtwnintr 23982 Inner transitivity law for betweenness. Left hand side of Theorem 3.5 of [Schwabhauser] p. 30. (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 ) ) ) 
 ->  ( ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. B ,  D >. )  ->  B  Btwn  <. A ,  C >. ) )
 
Theorembtwnexch3 23983 Exchange the first endpoint in betweenness. Left hand side of Theorem 3.6 of [Schwabhauser] p. 30. (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 ) ) ) 
 ->  ( ( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  D >. )  ->  C  Btwn  <. B ,  D >. ) )
 
Theorembtwnexch3and 23984 Deduction form of btwnexch3 23983. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  B  Btwn  <. A ,  C >. )   &    |-  ( ( ph  /\ 
 ps )  ->  C  Btwn  <. A ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  C  Btwn  <. B ,  D >. )
 
Theorembtwnouttr2 23985 Outer transitivity law for betweenness. Left hand side of Theorem 3.1 of [Schwabhauser] p. 30. (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 ) ) ) 
 ->  ( ( B  =/=  C 
 /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) 
 ->  C  Btwn  <. A ,  D >. ) )
 
Theorembtwnexch2 23986 Exchange the outer point of two betweenness statements. Right hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 14-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. B ,  D >. )  ->  C  Btwn  <. A ,  D >. ) )
 
Theorembtwnouttr 23987 Outer transitivity law for betweenness. Right hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 14-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( B  =/=  C 
 /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) 
 ->  B  Btwn  <. A ,  D >. ) )
 
Theorembtwnexch 23988 Outer transitivity law for betweenness. Right hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 24-Sep-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  D >. )  ->  B  Btwn  <. A ,  D >. ) )
 
Theorembtwnexchand 23989 Deduction form of btwnexch 23988. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  B  Btwn  <. A ,  C >. )   &    |-  ( ( ph  /\ 
 ps )  ->  C  Btwn  <. A ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  B  Btwn  <. A ,  D >. )
 
Theorembtwndiff 23990* There is always a  c distinct from  B such that  B lies between  A and  c. Theorem 3.14 of [Schwabhauser] p. 32. (Contributed by Scott Fenton, 24-Sep-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  E. c  e.  ( EE `  N ) ( B  Btwn  <. A ,  c >.  /\  B  =/=  c ) )
 
Theoremtrisegint 23991* A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Scott Fenton, 24-Sep-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) 
 /\  P  e.  ( EE `  N ) ) )  ->  ( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) 
 ->  E. q  e.  ( EE `  N ) ( q  Btwn  <. P ,  C >.  /\  q  Btwn  <. B ,  E >. ) ) )
 
16.7.34  Segment Transportation
 
Syntaxctransport 23992 Declare the syntax for the segment transport function.
 class TransportTo
 
Definitiondf-transport 23993* Define the segment transport function. See fvtransport 23995 for an explanation of the function. (Contributed by Scott Fenton, 18-Oct-2013.)
 |- TransportTo  =  { <.
 <. p ,  q >. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( ( EE
 `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n )  X.  ( EE `  n ) ) 
 /\  ( 1st `  q
 )  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  /\  <. ( 2nd `  q
 ) ,  r >.Cgr p ) ) ) }
 
Theoremfuntransport 23994 The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun TransportTo
 
Theoremfvtransport 23995* Calculate the value of the TransportTo function. This function takes four points,  A through  D, where  C and  D are distinct. It then returns the point that extends  C D by the length of  A B. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) 
 /\  C  =/=  D ) )  ->  ( <. A ,  B >.TransportTo <. C ,  D >. )  =  (
 iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C ,  r >.  /\  <. D ,  r >.Cgr <. A ,  B >. ) ) )
 
Theoremtransportcl 23996 Closure law for segment transport. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) 
 /\  C  =/=  D ) )  ->  ( <. A ,  B >.TransportTo <. C ,  D >. )  e.  ( EE `  N ) )
 
Theoremtransportprops 23997 Calculate the defining properties of the transport function (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) 
 /\  C  =/=  D ) )  ->  ( D 
 Btwn  <. C ,  ( <. A ,  B >.TransportTo <. C ,  D >. ) >.  /\ 
 <. D ,  ( <. A ,  B >.TransportTo <. C ,  D >. ) >.Cgr <. A ,  B >. ) )
 
16.7.35  Properties relating betweenness and congruence
 
Syntaxcifs 23998 Declare the syntax for the inner five segment predicate.
 class  InnerFiveSeg
 
Syntaxccgr3 23999 Declare the syntax for the three place congruence predicate.
 class Cgr3
 
Syntaxccolin 24000 Declare the syntax for the colinearity predicate.
 class  Colinear
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