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Theorem List for Metamath Proof Explorer - 25801-25900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxlowdimlem17 25801 Lemma for axlowdim 25804. 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 25802* 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 25803. (Contributed by Scott Fenton, 22-Apr-2013.)
 |-  ( N  e.  NN  ->  E. x  e.  ( EE
 `  N ) E. y  e.  ( EE `  N ) x  =/=  y )
 
Theoremaxlowdim2 25803* 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 25804* 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 25805* Lemma for axeuclid 25806. 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 25806* 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 25807* Lemma for axcont 25819. 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 25808* Lemma for axcont 25819. The idea here is to set up a mapping  F that will allow us to transfer dedekind 25140 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 25809* Lemma for axcont 25819. 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 25810* Lemma for axcont 25819. 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 25811* Lemma for axcont 25819. 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 25812* Lemma for axcont 25819. 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 25813* Lemma for axcont 25819. 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 25814* Lemma for axcont 25819. 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 25815* Lemma for axcont 25819. 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 25816* Lemma for axcont 25819. 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 25817* Lemma for axcont 25819. Eliminate the hypotheses from axcontlem10 25816. (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 25818* Lemma for axcont 25819. 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 25819* 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 >. )
 
19.7.41  Congruence properties
 
Syntaxcofs 25820 Declare the syntax for the outer five segment configuration.
 class  OuterFiveSeg
 
Definitiondf-ofs 25821* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 25781). See brofs 25843 and 5segofs 25844 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 25822 Deduction form of axcgrrflx 25757. (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 25823 Deduction form of axcgrtr 25758. (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 25824 Deduction form of axcgrtr 25758. (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 25825 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 25826 Deduction form of cgrrflx 25825. (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 25827 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 25828 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 25829 Deduction form of cgrcom 25828. (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 25830 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 25831 Deduction form of cgrtr 25830. (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 25832 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 25833 Deduction form of cgrtr3 25832. (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 25834 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 25835 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 25836 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 25837 Deduction form of cgrcoml 25834. (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 25838 Deduction form of cgrcoml 25834. (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 25839 Deduction form of cgrcomlr 25836. (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 25840 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 25841 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 25842 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 25843 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 25844 Rephrase ax5seg 25781 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 25845 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 25846 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 25847 Deduction form of cgrextend 25846. (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 25848 Two points that satsify the conclusion of axsegcon 25770 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 25849* Existential uniqueness version of segconeq 25848. (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 >. ) )
 
19.7.42  Betweenness properties
 
Theorembtwntriv2 25850 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 25851 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 25852 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 25853 Deduction form of btwncom 25852. (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 25854 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 25855 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 25856 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 25857 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 25858 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 25859 Deduction form of btwnexch3 25858. (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 25860 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 25861 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 25862 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 25863 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 25864 Deduction form of btwnexch 25863. (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 25865* 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 25866* 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 >. ) ) )
 
19.7.43  Segment Transportation
 
Syntaxctransport 25867 Declare the syntax for the segment transport function.
 class TransportTo
 
Definitiondf-transport 25868* Define the segment transport function. See fvtransport 25870 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 25869 The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun TransportTo
 
Theoremfvtransport 25870* 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 25871 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 25872 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 >. ) )
 
19.7.44  Properties relating betweenness and congruence
 
Syntaxcifs 25873 Declare the syntax for the inner five segment predicate.
 class  InnerFiveSeg
 
Syntaxccgr3 25874 Declare the syntax for the three place congruence predicate.
 class Cgr3
 
Syntaxccolin 25875 Declare the syntax for the colinearity predicate.
 class  Colinear
 
Syntaxcfs 25876 Declare the syntax for the five segment predicate.
 class  FiveSeg
 
Definitiondf-ifs 25877* The inner five segment configuration is an abbreviation for another congruence condition. See brifs 25881 and ifscgr 25882 for how it is used. Definition 4.1 of [Schwabhauser] p. 34. (Contributed by Scott Fenton, 26-Sep-2013.)
 |-  InnerFiveSeg  =  { <. 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 ,  c >.Cgr <. x ,  z >.  /\ 
 <. b ,  c >.Cgr <.
 y ,  z >. ) 
 /\  ( <. a ,  d >.Cgr <. x ,  w >.  /\  <. c ,  d >.Cgr
 <. z ,  w >. ) ) ) }
 
Definitiondf-cgr3 25878* The three place congruence predicate. This is an abbreviation for saying that all three pair in a triple are congruent with each other. Three place form of Definition 4.4 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
 |- Cgr3  =  { <. 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. e  e.  ( EE `  n ) E. f  e.  ( EE `  n ) ( p  = 
 <. a ,  <. b ,  c >. >.  /\  q  =  <. d ,  <. e ,  f >. >.  /\  ( <. a ,  b >.Cgr <. d ,  e >.  /\  <. a ,  c >.Cgr <. d ,  f >.  /\  <. b ,  c >.Cgr
 <. e ,  f >. ) ) }
 
Definitiondf-colinear 25879* The colinearity predicate states that the three points in its arguments sit on one line. Definition 4.10 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 25-Oct-2013.)
 |-  Colinear  =  `' { <. <. b ,  c >. ,  a >.  |  E. n  e.  NN  (
 ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n ) )  /\  ( a 
 Btwn  <. b ,  c >.  \/  b  Btwn  <. c ,  a >.  \/  c  Btwn  <. a ,  b >. ) ) }
 
Definitiondf-fs 25880* The general five segment configuration is a generalization of the outer and inner five segment configurations. See brfs 25917 and fscgr 25918 for its use. Definition 4.15 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  FiveSeg  =  { <. 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 >. >.  /\  ( a  Colinear  <. b ,  c >.  /\  <. a , 
 <. b ,  c >. >.Cgr3 <. x ,  <. y ,  z >. >.  /\  ( <. a ,  d >.Cgr <. x ,  w >.  /\  <. b ,  d >.Cgr <. y ,  w >. ) ) ) }
 
Theorembrifs 25881 Binary relationship form of the inner five segment predicate. (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 >. >.  InnerFiveSeg  <. <. E ,  F >. , 
 <. G ,  H >. >.  <->  ( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. ) 
 /\  ( <. A ,  C >.Cgr <. E ,  G >.  /\  <. B ,  C >.Cgr
 <. F ,  G >. ) 
 /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. C ,  D >.Cgr
 <. G ,  H >. ) ) ) )
 
Theoremifscgr 25882 Inner five segment congruence. Take two triangles,  A D C and  E H G, with 
B between  A and  C and  F between  E and  G. If the other components of the triangles are congruent, then so are  B D and  F H. Theorem 4.2 of [Schwabhauser] p. 34. (Contributed by Scott Fenton, 27-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 >. >.  InnerFiveSeg  <. <. E ,  F >. , 
 <. G ,  H >. >.  -> 
 <. B ,  D >.Cgr <. F ,  H >. ) )
 
Theoremcgrsub 25883 Removing identical parts from the end of a line segment preserves congruence. Theorem 4.3 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-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 ) ) )  ->  ( (
 ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. ) 
 /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
 <. E ,  F >. ) )  ->  <. A ,  B >.Cgr <. D ,  E >. ) )
 
Theorembrcgr3 25884 Binary relationship form of the three-place congruence predicate. (Contributed by Scott Fenton, 4-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 ,  C >.
 >.Cgr3 <. D ,  <. E ,  F >. >.  <->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr
 <. D ,  F >.  /\ 
 <. B ,  C >.Cgr <. E ,  F >. ) ) )
 
Theoremcgr3permute3 25885 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-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 ,  C >.
 >.Cgr3 <. D ,  <. E ,  F >. >.  <->  <. B ,  <. C ,  A >. >.Cgr3 <. E ,  <. F ,  D >. >.
 ) )
 
Theoremcgr3permute1 25886 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-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 ,  C >.
 >.Cgr3 <. D ,  <. E ,  F >. >.  <->  <. A ,  <. C ,  B >. >.Cgr3 <. D ,  <. F ,  E >. >.
 ) )
 
Theoremcgr3permute2 25887 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-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 ,  C >.
 >.Cgr3 <. D ,  <. E ,  F >. >.  <->  <. B ,  <. A ,  C >. >.Cgr3 <. E ,  <. D ,  F >. >.
 ) )
 
Theoremcgr3permute4 25888 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-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 ,  C >.
 >.Cgr3 <. D ,  <. E ,  F >. >.  <->  <. C ,  <. A ,  B >. >.Cgr3 <. F ,  <. D ,  E >. >.
 ) )
 
Theoremcgr3permute5 25889 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-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 ,  C >.
 >.Cgr3 <. D ,  <. E ,  F >. >.  <->  <. C ,  <. B ,  A >. >.Cgr3 <. F ,  <. E ,  D >. >.
 ) )
 
Theoremcgr3tr4 25890 Transitivity law for three-place congruence. (Contributed by Scott Fenton, 5-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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N ) 
 /\  I  e.  ( EE `  N ) ) ) )  ->  (
 ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. G ,  <. H ,  I >. >.
 )  ->  <. D ,  <. E ,  F >. >.Cgr3 <. G ,  <. H ,  I >. >. ) )
 
Theoremcgr3com 25891 Commutativity law for three-place congruence. (Contributed by Scott Fenton, 5-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 ,  C >.
 >.Cgr3 <. D ,  <. E ,  F >. >.  <->  <. D ,  <. E ,  F >. >.Cgr3 <. A ,  <. B ,  C >. >.
 ) )
 
Theoremcgr3rflx 25892 Identity law for three-place congruence. (Contributed by Scott Fenton, 6-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >. )
 
Theoremcgrxfr 25893* A line segment can be divided at the same place as a congruent line segment is divided. Theorem 4.5 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )  ->  (
 ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. )  ->  E. e  e.  ( EE `  N ) ( e  Btwn  <. D ,  F >.  /\ 
 <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )
 
Theorembtwnxfr 25894 A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 4-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 ) ) )  ->  ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.
 )  ->  E  Btwn  <. D ,  F >. ) )
 
Theoremcolinrel 25895 Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Rel  Colinear
 
Theorembrcolinear2 25896* Alternate colinearity binary relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( Q  e.  V  /\  R  e.  W ) 
 ->  ( P  Colinear  <. Q ,  R >. 
 <-> 
 E. n  e.  NN  ( ( P  e.  ( EE `  n ) 
 /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) )  /\  ( P  Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) ) ) )
 
Theorembrcolinear 25897 The binary relationship form of the colinearity predicate. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Colinear  <. B ,  C >. 
 <->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )
 
Theoremcolinearex 25898 The colinear predicate exists. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Colinear  e.  _V
 
Theoremcolineardim1 25899 If  A is colinear with  B and  C, then  A is in the same space as  B. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE
 `  N )  /\  C  e.  W )
 )  ->  ( A  Colinear  <. B ,  C >.  ->  A  e.  ( EE `  N ) ) )
 
Theoremcolinearperm1 25900 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Colinear  <. B ,  C >. 
 <->  A  Colinear  <. C ,  B >. ) )
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