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Theorem List for Metamath Proof Explorer - 26101-26200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisconcl5a 26101* Two distinct lines intersect in at most one point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  L  =  (PLines `  F )   &    |-  P  =  (PPoints `  F )   &    |-  ( ph  ->  F  e. Ig )   &    |-  ( ph  ->  L1  e.  L )   &    |-  ( ph  ->  L 2  e.  L )   &    |-  ( ph  ->  L1  =/=  L 2 )   =>    |-  ( ph  ->  E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
 
Theoremisconcl5ab 26102* Two distinct lines intersect in at most one point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  L  =  (PLines `  F )   &    |-  P  =  (PPoints `  F )   &    |-  ( ph  ->  F  e. Ig )   &    |-  ( ph  ->  L1  e.  L )   &    |-  ( ph  ->  L 2  e.  L )   &    |-  ( ph  ->  L1  =/=  L 2 )   =>    |-  ( ph  ->  E* p ( p  e.  L1  /\  p  e.  L 2 ) )
 
Theoremisconcl6a 26103* Two distinct non-parallel lines intersect in one and only point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  L  =  (PLines `  F )   &    |-  P  =  (PPoints `  F )   &    |-  ( ph  ->  F  e. Ig )   &    |-  ( ph  ->  L1  e.  L )   &    |-  ( ph  ->  L 2  e.  L )   &    |-  ( ph  ->  L1  =/=  L 2 )   &    |-  ( ph  ->  ( L1  i^i  L 2 )  =/=  (/) )   =>    |-  ( ph  ->  E! p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
 
Theoremisconcl6ab 26104* Two distinct non-parallel lines intersect in one and only point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  L  =  (PLines `  F )   &    |-  P  =  (PPoints `  F )   &    |-  ( ph  ->  F  e. Ig )   &    |-  ( ph  ->  L1  e.  L )   &    |-  ( ph  ->  L 2  e.  L )   &    |-  ( ph  ->  L1  =/=  L 2 )   &    |-  ( ph  ->  ( L1  i^i  L 2 )  =/=  (/) )   =>    |-  ( ph  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) )
 
Theoremisconcl7a 26105 Two distinct non-parallel lines intersect in one and only point. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  L  =  (PLines `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  L1  e.  L )   &    |-  ( ph  ->  L 2  e.  L )   &    |-  ( ph  ->  L1  =/=  L 2 )   &    |-  ( ph  ->  X  e.  ( L1 
 i^i  L 2 ) )   =>    |-  ( ph  ->  ( L1  i^i  L 2 )  =  { X } )
 
Syntaxcbtw 26106 Extend class notation with the betweenness relation.
 class btw
 
Syntaxcibg 26107 Extend class notation with the class of all planar incidence betweenness geometries.
 class Ibg
 
Definitiondf-btw 26108 Definition of btw. (Contributed by FL, 1-Apr-2016.)
 |- btw  =  +g
 
Definitiondf-ibg2 26109* Definition of a geometry that can build on the axioms of incidence and betweenness. Axioms B-1, B-2, B-3, B-4 of [AitkenIBG] p. 3-4. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
 |- Ibg  =  {
 f  e. Ig  |  [. (PPoints `  f )  /  g ]. [. (PLines `  f
 )  /  h ]. [. (coln `  f )  /  d ]. [. (btw `  f
 )  /  e ]. A. p  e.  g  A. q  e.  g  (
 ( p  =/=  q  ->  E. a  e.  g  E. b  e.  g  E. c  e.  g  ( p  e.  (
 a e q ) 
 /\  b  e.  ( p e q ) 
 /\  q  e.  ( p e c ) ) )  /\  A. r  e.  g  (
 ( ( { p ,  q ,  r }  e.  d  /\  ( p  =/=  q  /\  p  =/=  r  /\  q  =/=  r ) )  ->  ( ( p  e.  ( q e r )  /\  q  e/  ( p e r ) 
 /\  r  e/  ( p e q ) )  \/  ( p 
 e/  ( q e r )  /\  q  e.  ( p e r )  /\  r  e/  ( p e q ) )  \/  ( p 
 e/  ( q e r )  /\  q  e/  ( p e r )  /\  r  e.  ( p e q ) ) ) ) 
 /\  ( q  e.  ( p e r )  ->  ( q  e.  ( r e p )  /\  { p ,  q ,  r }  e.  d  /\  ( p  =/=  q  /\  p  =/=  r  /\  q  =/=  r ) ) ) 
 /\  A. l  e.  h  ( ( p  e/  l  /\  q  e/  l  /\  r  e/  l ) 
 ->  ( ( ( ( ( p e q )  i^i  l )  =  (/)  /\  ( (
 q e r )  i^i  l )  =  (/) )  ->  ( ( p e r )  i^i  l )  =  (/) )  /\  ( ( ( ( p e q )  i^i  l
 )  =/=  (/)  /\  (
 ( q e r )  i^i  l )  =/=  (/) )  ->  (
 ( p e r )  i^i  l )  =  (/) ) ) ) ) ) }
 
Theoremisibg2 26110* The predicate "is an incidence-betweenness geometry".

B-1 If  q is between  p and  r then it is between  r and  p and  p,  q,  r are collinear and distinct.

B-2 If  p and  q are distinct, then there are points  a,  b,  c such that  a is before  p and  q,  b is between  p and  q,  c is after  p and  q.

B-3 If three points  p,  q,  r are collinear and distinct then exactly one of the followings occurs:  p is between  q and  r,  q is between  p and  r,  r is between  p and  q.

B-4 "Being on the same side" is a transitive relation. If  p and  q are not on the same side of  l and  q and  r are not on the same side of  l then  p and  r are on the same side of  l. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)

 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  B  =  (btw `  G )   &    |-  C  =  (coln `  G )   =>    |-  ( G  e. Ibg  <->  ( G  e. Ig  /\ 
 A. p  e.  P  A. q  e.  P  ( ( p  =/=  q  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  ( p  e.  (
 a B q ) 
 /\  b  e.  ( p B q )  /\  q  e.  ( p B c ) ) )  /\  A. r  e.  P  ( ( ( { p ,  q ,  r }  e.  C  /\  ( p  =/=  q  /\  p  =/=  r  /\  q  =/=  r
 ) )  ->  (
 ( p  e.  (
 q B r ) 
 /\  q  e/  ( p B r )  /\  r  e/  ( p B q ) )  \/  ( p  e/  (
 q B r ) 
 /\  q  e.  ( p B r )  /\  r  e/  ( p B q ) )  \/  ( p  e/  (
 q B r ) 
 /\  q  e/  ( p B r )  /\  r  e.  ( p B q ) ) ) )  /\  (
 ( q  e.  ( p B r )  ->  ( q  e.  (
 r B p ) 
 /\  { p ,  q ,  r }  e.  C  /\  ( p  =/=  q  /\  p  =/=  r  /\  q  =/=  r
 ) ) )  /\  A. l  e.  L  ( ( p  e/  l  /\  q  e/  l  /\  r  e/  l )  ->  ( ( ( ( ( p B q )  i^i  l )  =  (/)  /\  ( (
 q B r )  i^i  l )  =  (/) )  ->  ( ( p B r )  i^i  l )  =  (/) )  /\  ( ( ( ( p B q )  i^i  l
 )  =/=  (/)  /\  (
 ( q B r )  i^i  l )  =/=  (/) )  ->  (
 ( p B r )  i^i  l )  =  (/) ) ) ) ) ) ) ) )
 
Theoremisibg1a 26111 An incidence-betweenness geometry is an incidence geometry. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
 |-  ( ph  ->  G  e. Ibg )   =>    |-  ( ph  ->  G  e. Ig )
 
Theoremisibg2aa 26112* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  B  =  (btw `  G )   &    |-  C  =  (coln `  G )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  M  e.  L )   &    |-  ( ph  ->  G  e. Ibg )   =>    |-  ( ph  ->  (
 ( X  =/=  Y  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  ( X  e.  (
 a B Y ) 
 /\  b  e.  ( X B Y )  /\  Y  e.  ( X B c ) ) )  /\  ( ( ( { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) )  ->  (
 ( X  e.  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X 
 e/  ( Y B Z )  /\  Y  e.  ( X B Z ) 
 /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z ) 
 /\  Y  e/  ( X B Z )  /\  Z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B Z )  ->  ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) ) )  /\  ( ( X  e/  M  /\  Y  e/  M  /\  Z  e/  M )  ->  (
 ( ( ( ( X B Y )  i^i  M )  =  (/)  /\  ( ( Y B Z )  i^i 
 M )  =  (/) )  ->  ( ( X B Z )  i^i 
 M )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  M )  =/=  (/)  /\  ( ( Y B Z )  i^i 
 M )  =/=  (/) )  ->  ( ( X B Z )  i^i  M )  =  (/) ) ) ) ) ) ) )
 
Theoremisibg2aalem1 26113* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   =>    |-  (
 ( X  e.  P  /\  Y  e.  P ) 
 ->  ( A. x  e.  P  A. y  e.  P  ( ( x  =/=  y  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  ( x  e.  ( a B y )  /\  b  e.  ( x B y )  /\  y  e.  ( x B c ) ) )  /\  A. z  e.  P  ( ( ( { x ,  y ,  z }  e.  C  /\  ( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  ->  ( ( x  e.  ( y B z )  /\  y  e/  ( x B z ) 
 /\  z  e/  ( x B y ) )  \/  ( x  e/  ( y B z )  /\  y  e.  ( x B z )  /\  z  e/  ( x B y ) )  \/  ( x 
 e/  ( y B z )  /\  y  e/  ( x B z )  /\  z  e.  ( x B y ) ) ) ) 
 /\  ( ( y  e.  ( x B z )  ->  (
 y  e.  ( z B x )  /\  { x ,  y ,  z }  e.  C  /\  ( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z
 ) ) )  /\  A. l  e.  L  ( ( x  e/  l  /\  y  e/  l  /\  z  e/  l )  ->  ( ( ( ( ( x B y )  i^i  l )  =  (/)  /\  ( (
 y B z )  i^i  l )  =  (/) )  ->  ( ( x B z )  i^i  l )  =  (/) )  /\  ( ( ( ( x B y )  i^i  l
 )  =/=  (/)  /\  (
 ( y B z )  i^i  l )  =/=  (/) )  ->  (
 ( x B z )  i^i  l )  =  (/) ) ) ) ) ) )  ->  ( ( X  =/=  Y 
 ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  ( X  e.  (
 a B Y ) 
 /\  b  e.  ( X B Y )  /\  Y  e.  ( X B c ) ) )  /\  A. z  e.  P  ( ( ( { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) ) 
 ->  ( ( X  e.  ( Y B z ) 
 /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) ) 
 /\  ( ( Y  e.  ( X B z )  ->  ( Y  e.  ( z B X )  /\  { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) ) )  /\  A. l  e.  L  ( ( X 
 e/  l  /\  Y  e/  l  /\  z  e/  l )  ->  ( ( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B z )  i^i  l )  =  (/) )  ->  (
 ( X B z )  i^i  l )  =  (/) )  /\  (
 ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  (
 ( Y B z )  i^i  l )  =/=  (/) )  ->  (
 ( X B z )  i^i  l )  =  (/) ) ) ) ) ) ) ) )
 
Theoremisibg2aalem2 26114* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
 |-  ( Z  e.  P  ->  (
 A. z  e.  P  ( ( ( { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) ) 
 ->  ( ( X  e.  ( Y B z ) 
 /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) ) 
 /\  ( ( Y  e.  ( X B z )  ->  ( Y  e.  ( z B X )  /\  { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) ) )  /\  A. l  e.  L  ( ( X 
 e/  l  /\  Y  e/  l  /\  z  e/  l )  ->  ( ( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B z )  i^i  l )  =  (/) )  ->  (
 ( X B z )  i^i  l )  =  (/) )  /\  (
 ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  (
 ( Y B z )  i^i  l )  =/=  (/) )  ->  (
 ( X B z )  i^i  l )  =  (/) ) ) ) ) )  ->  (
 ( ( { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) )  ->  (
 ( X  e.  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X 
 e/  ( Y B Z )  /\  Y  e.  ( X B Z ) 
 /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z ) 
 /\  Y  e/  ( X B Z )  /\  Z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B Z )  ->  ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) ) )  /\  A. l  e.  L  ( ( X 
 e/  l  /\  Y  e/  l  /\  Z  e/  l )  ->  ( ( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l )  =  (/) )  ->  (
 ( X B Z )  i^i  l )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) ) ) ) ) ) ) )
 
Theoremisibg2aalem3 26115* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
 |-  ( M  e.  L  ->  (
 A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  Z  e/  l ) 
 ->  ( ( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l )  =  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) ) ) ) 
 ->  ( ( X  e/  M  /\  Y  e/  M  /\  Z  e/  M ) 
 ->  ( ( ( ( ( X B Y )  i^i  M )  =  (/)  /\  ( ( Y B Z )  i^i 
 M )  =  (/) )  ->  ( ( X B Z )  i^i 
 M )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  M )  =/=  (/)  /\  ( ( Y B Z )  i^i 
 M )  =/=  (/) )  ->  ( ( X B Z )  i^i  M )  =  (/) ) ) ) ) )
 
Theoremisib2g1a1 26116 If  Y is between  X and  Z, it is between  Z and  X (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   =>    |-  ( ph  ->  Y  e.  ( Z B X ) )
 
Theoremisibg1a2 26117 If  Y is between  X and  Z, then  X,  Y,  Z are collinear . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  C  =  (coln `  G )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   &    |-  ( ph  ->  Z  e.  P )   =>    |-  ( ph  ->  { X ,  Y ,  Z }  e.  C )
 
Theoremisibg2a 26118* Two distinct points have a point before, between and after them. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  ( X  e.  ( a B Y )  /\  b  e.  ( X B Y )  /\  Y  e.  ( X B c ) ) )
 
Theoremisibg2a1 26119* Two distinct points  X,  Y have a point before them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  E. a  e.  P  X  e.  ( a B Y ) )
 
Theoremisibg2a2 26120* Two distinct points  X,  Y have a point between them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  E. b  e.  P  b  e.  ( X B Y ) )
 
Theoremisibg2a3 26121* Two distinct points  X,  Y have a point after them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  E. c  e.  P  Y  e.  ( X B c ) )
 
Theoremisibg1a3a 26122 If  Y is between  X and  Z, then  X and 
Y, are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   =>    |-  ( ph  ->  X  =/=  Y )
 
Theoremisibg1spa 26123 If  Y is between  X and  Z, then  X and 
Z, are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   =>    |-  ( ph  ->  X  =/=  Z )
 
Theoremisibg1a5a 26124 If  Y is between  X and  Z, then  Y and 
Z, are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   =>    |-  ( ph  ->  Y  =/=  Z )
 
Theoremisibg1a6 26125 If  Y is between  X and  Z, it belongs to the line XZ (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   &    |-  M  =  ( line `  G )   =>    |-  ( ph  ->  Y  e.  ( X M Z ) )
 
Theoremisibg1a7 26126 If  Y is between  X and  Z,  X belongs to the line YZ (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   &    |-  M  =  ( line `  G )   =>    |-  ( ph  ->  X  e.  ( Y M Z ) )
 
Theoremisibg1a8 26127 If  Y is between  X and  Z,  Z belongs to the line XY (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   &    |-  M  =  ( line `  G )   =>    |-  ( ph  ->  Z  e.  ( X M Y ) )
 
Theorembsstr 26128 Being on the same side is a transitive relation. (For my private use only. Don't use.) (Contributed by FL, 10-Jul-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  ( ph  ->  X  e.  ( P  \  M ) )   &    |-  ( ph  ->  Y  e.  ( P  \  M ) )   &    |-  ( ph  ->  Z  e.  ( P  \  M ) )   &    |-  ( ph  ->  ( ( X B Y )  i^i 
 M )  =  (/) )   &    |-  ( ph  ->  (
 ( Y B Z )  i^i  M )  =  (/) )   =>    |-  ( ph  ->  (
 ( X B Z )  i^i  M )  =  (/) )
 
Theoremnbssntr 26129 IF  X and  Y are not on the same side, and  Y and  Z are not on the same side then 
X and  Z are on the same side. (For my private use only. Don't use.) (Contributed by FL, 10-Jul-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  ( ph  ->  X  e.  ( P  \  M ) )   &    |-  ( ph  ->  Y  e.  ( P  \  M ) )   &    |-  ( ph  ->  Z  e.  ( P  \  M ) )   &    |-  ( ph  ->  ( ( X B Y )  i^i 
 M )  =/=  (/) )   &    |-  ( ph  ->  ( ( Y B Z )  i^i 
 M )  =/=  (/) )   =>    |-  ( ph  ->  ( ( X B Z )  i^i  M )  =  (/) )
 
Syntaxcseg 26130 Extend class notation with the segment symbol.
 class  seg
 
Definitiondf-seg2 26131* Definition of the segment xy degenerated or not. Definition 8 of [AitkenIBG] p. 4. (For my private use only. Don't use.) (Contributed by FL, 28-Feb-2016.)
 |-  seg  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f
 ) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  f )  |  ( z  e.  ( x (btw `  f )
 y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) )
 
Theoremsgplpte21 26132* The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  ( X S Y )  =  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } )
 
Theoremsgplpte21a 26133* The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  A. z
 ( z  e.  ( X S Y )  <->  ( z  e.  P  /\  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) ) ) )
 
Theoremsgplpte21b 26134 The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Z  e.  _V )   =>    |-  ( ph  ->  ( Z  e.  ( X S Y )  <->  ( Z  e.  P  /\  ( Z  e.  ( X B Y )  \/  Z  =  X  \/  Z  =  Y ) ) ) )
 
Theoremsgplpte21c 26135 The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  ( Z  e.  ( X S Y )  ->  ( Z  e.  P  /\  ( Z  e.  ( X B Y )  \/  Z  =  X  \/  Z  =  Y )
 ) ) )
 
Theoremsgplpte21d 26136 The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  (
 ( Z  e.  P  /\  ( Z  e.  ( X B Y )  \/  Z  =  X  \/  Z  =  Y )
 )  ->  Z  e.  ( X S Y ) ) )
 
Theoremsgplpte21e 26137 The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  ( Z  e.  ( X S Y )  <->  ( Z  e.  P  /\  ( Z  e.  ( X B Y )  \/  Z  =  X  \/  Z  =  Y ) ) ) )
 
Theoremsgplpte22 26138 The predicate "is a degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   =>    |-  ( ph  ->  ( X S X )  =  { X }
 )
 
Theoremsgplpte21d1 26139 The extremities belong to a segment. (For my private use only. Don't use.) (Contributed by FL, 29-Jul-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   =>    |-  ( ph  ->  X  e.  ( X S Y ) )
 
Theoremsgplpte21d2 26140 The extremities belong to a segment. (For my private use only. Don't use.) (Contributed by FL, 29-Jul-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   =>    |-  ( ph  ->  Y  e.  ( X S Y ) )
 
Theoremsegline 26141 A segment is a part of a line. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  Y  e.  P )   =>    |-  ( ph  ->  ( X S Y )  C_  ( X M Y ) )
 
Theorempxysxy 26142 Points between  X and  Y belong to the segment XY. (For my private use only. Don't use.) (Contributed by FL, 17-Jul-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  (
 ( X B Y )  i^i  P )  C_  ( X S Y ) )
 
Theoremlppotoslem 26143 To show that a point doesn't belong to a line  M 2. (For my private use only. Don't use.) (Contributed by FL, 14-Jul-2016.)
 |-  ( ph  ->  ( M1  i^i  M 2 )  =  { X } )   &    |-  ( ph  ->  Y  e.  M1 )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  -.  Y  e.  M 2
 )
 
Theoremlppotos 26144* Given a line  M and a point  X not on this line. There exists a point on the other side of the line. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  ( P  \  M ) )   &    |-  ( ph  ->  M  e.  L )   =>    |-  ( ph  ->  E. y  e.  ( P  \  M ) ( ( X S y )  i^i 
 M )  =/=  (/) )
 
Theoremxsyysx 26145 The segments xy and yx are equal. (For my private use only. Don't use.) (Contributed by FL, 17-Jul-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   =>    |-  ( ph  ->  ( X S Y )  =  ( Y S X ) )
 
Theorembsstrs 26146 Being on the same side is a transitive relation. Segment version of bsstr 26128. (For my private use only. Don't use.) (Contributed by FL, 14-Jul-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  ( ph  ->  X  e.  ( P  \  M ) )   &    |-  ( ph  ->  Y  e.  ( P  \  M ) )   &    |-  ( ph  ->  Z  e.  ( P  \  M ) )   &    |-  ( ph  ->  ( ( X S Y )  i^i 
 M )  =  (/) )   &    |-  ( ph  ->  (
 ( Y S Z )  i^i  M )  =  (/) )   =>    |-  ( ph  ->  (
 ( X S Z )  i^i  M )  =  (/) )
 
Theoremnbssntrs 26147 IF  X and  Y are not on the same side, and  Y and  Z are not on the same side then 
X and  Z are on the same side. (For my private use only. Don't use.) (Contributed by FL, 14-Jul-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  ( ph  ->  X  e.  ( P  \  M ) )   &    |-  ( ph  ->  Y  e.  ( P  \  M ) )   &    |-  ( ph  ->  Z  e.  ( P  \  M ) )   &    |-  ( ph  ->  ( ( X S Y )  i^i 
 M )  =/=  (/) )   &    |-  ( ph  ->  ( ( Y S Z )  i^i 
 M )  =/=  (/) )   =>    |-  ( ph  ->  ( ( X S Z )  i^i  M )  =  (/) )
 
SyntaxcSeg 26148 Extend class notation with the non-degenerated segment symbol.
 class Segments
 
Definitiondf-Seg 26149 The non-degenerated segments. (For my private use only. Don't use.) (Contributed by FL, 7-Mar-2016.)
 |- Segments  =  ( f  e. Ibg  |->  ( ( seg `  f ) " ( ( (PPoints `  f
 )  X.  (PPoints `  f
 ) )  \  _I  ) ) )
 
Theoremnds 26150 The non-degenerated segments. (For my private use only. Don't use.) (Contributed by FL, 7-Mar-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   =>    |-  ( G  e. Ibg  ->  (Segments `  G )  =  ( S " ( ( P  X.  P ) 
 \  _I  ) ) )
 
Syntaxcray2 26151 Extend class notation with the ray symbol.
 class ray
 
Definitiondf-ray2 26152* Definition of the ray xy degenerated or not. Definition 11 of [AitkenIBG] p. 4. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
 |- ray  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f
 ) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  ( ( x ( seg `  f
 ) y )  u. 
 { z  e.  (PPoints `  f )  |  y  e.  ( x (btw `  f ) z ) } ) ,  { x } ) ) )
 
Theoremisray2 26153 A degenerated ray. (For my private use only. Don't use.) (Contributed by FL, 13-Apr-2016.)
 |-  P  =  (PPoints `  G )   &    |-  R  =  (ray `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   =>    |-  ( ph  ->  ( X R X )  =  { X } )
 
Theoremisray 26154* A non-degenerated ray. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  R  =  (ray `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  ( X R Y )  =  ( ( X S Y )  u. 
 { z  e.  P  |  Y  e.  ( X B z ) }
 ) )
 
Theoremsegray 26155 A segment is a part of a ray. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  R  =  (ray `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   =>    |-  ( ph  ->  ( X S Y )  C_  ( X R Y ) )
 
Theoremrayline 26156 A ray is a part of a line. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  ( line `  G )   &    |-  R  =  (ray `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   =>    |-  ( ph  ->  ( X R Y )  C_  ( X L Y ) )
 
Syntaxcangle 26157 Extend class notation with the angle symbol.
 class angle
 
Definitiondf-angle 26158* Definition of an angle. Definition 17 of [AitkenIBG] p. 10. The angles can't be degenerated. Contrary to the concept of degenerated line or segment, the concept of degenerated angle no longer simplifies the wording. (For my private use only. Don't use.) (Contributed by FL, 7-Apr-2016.)
 |- angle  =  ( g  e. Ibg  |->  ( w  e.  ( (PPoints `  g
 )dWords 3 )  |->  ( ( ( w `  2 ) (ray `  g ) ( w `
  1 ) )  u.  ( ( w `
  2 ) (ray `  g ) ( w `
  3 ) ) ) ) )
 
Syntaxctriangle 26159 Extend class notation with the triangle symbol.
 class triangle
 
Definitiondf-triangle 26160* Definition of a triangle, degenerated or not. Definition 23 of [AitkenIBG] p. 15. (For my private use only. Don't use.) (Contributed by FL, 7-Apr-2016.)
 |- triangle  =  ( g  e. Ibg  |->  ( w  e.  ( (PPoints `  g
 )dWords 3 )  |->  ( ( ( ( w `
  1 ) ( seg `  g )
 ( w `  2
 ) )  u.  (
 ( w `  2
 ) ( seg `  g
 ) ( w `  3 ) ) )  u.  ( ( w `
  3 ) ( seg `  g )
 ( w `  1
 ) ) ) ) )
 
Syntaxctrcng 26161 Extend class notation with triangle congruence.
 class trcng
 
Syntaxcsas 26162 Extend class notation with the same side relation.
 class ss
 
Definitiondf-sside 26163* Two points not on  l are on the same side of  l if the segment xy doesn't intersect  l. Definition 10 of [AitkenIBG] p. 4 (For my private use only. Don't use.) (Contributed by FL, 26-May-2016.)
 |- ss  =  ( g  e. Ibg  |->  ( l  e.  (PLines `  g
 )  |->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  g
 )  \  l )  /\  y  e.  (
 (PPoints `  g )  \  l )  /\  ( ( x ( seg `  g
 ) y )  i^i  l )  =  (/) ) } ) )
 
Theoremisside0 26164* The predicate "Being on the same side of  L " (For my private use only. Don't use.) (Contributed by FL, 19-Jun-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  .~  =  ( (ss `  G ) `  M )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  S  =  ( seg `  G )   =>    |-  ( ph  ->  .~  =  { <. x ,  y >.  |  ( x  e.  ( P  \  M )  /\  y  e.  ( P  \  M )  /\  ( ( x S y )  i^i  M )  =  (/) ) }
 )
 
Theoremisside1 26165 The predicate "Being on the same side of  L " (For my private use only. Don't use.) (Contributed by FL, 19-Jun-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  .~  =  ( (ss `  G ) `  M )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .~  Y  <->  ( X  e.  ( P  \  M ) 
 /\  Y  e.  ( P  \  M )  /\  ( ( X S Y )  i^i  M )  =  (/) ) ) )
 
Theoremisside 26166 The predicate "Being on the same side of  L " (For my private use only. Don't use.) (Contributed by FL, 19-Jun-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  .~  =  ( (ss `  G ) `  M )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  X  e.  ( P  \  M ) )   &    |-  ( ph  ->  Y  e.  ( P  \  M ) )   =>    |-  ( ph  ->  ( X  .~  Y  <->  ( ( X S Y )  i^i 
 M )  =  (/) ) )
 
Theorembosser 26167 "Being on the same side of  M " is an equivalence relation among points that are not on  M. (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  .~  =  ( (ss `  G ) `  M )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   =>    |-  ( ph  ->  .~  Er  ( P  \  M ) )
 
Theorempdiveql 26168 The plane is divided into two equivalence classes by a line  M. (For my private use only. Don't use.) (Contributed by FL, 14-Jul-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  .~  =  ( (ss `  G ) `  M )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  ( ph  ->  A  e.  ( P  \  M ) )   =>    |-  ( ph  ->  ( B  e.  ( ( P  \  M ) /.  .~  )  <->  ( B  =  [ A ]  .~  \/  B  =  ( ( P  \  M )  \  [ A ]  .~  )
 ) ) )
 
Theoremhpd 26169 Halfplanes are distinct. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  .~  =  ( (ss `  G ) `  M )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  ( ph  ->  A  e.  ( P  \  M ) )   =>    |-  ( ph  ->  [ A ]  .~  =/=  ( ( P  \  M ) 
 \  [ A ]  .~  ) )
 
Syntaxchalfp 26170 Extend class notation with the Halfplane symbol.
 class Halfplane
 
Definitiondf-halfplane 26171* Returns the halplanes delimited by  l. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |- Halfplane  =  ( g  e. Ibg  |->  ( l  e.  (PLines `  g
 )  |->  ( ( (PPoints `  g )  \  l
 ) /. ( (ss `  g ) `  l
 ) ) ) )
 
Theoremaishp 26172 The half planes delimited by  M. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  P  =  (PPoints `  G )   &    |- 
 .~  =  ( (ss
 `  G ) `  M )   &    |-  L  =  (PLines `  G )   =>    |-  ( ph  ->  (
 (Halfplane `  G ) `  M )  =  (
 ( P  \  M ) /.  .~  ) )
 
Theoremabhp 26173* The half planes delimited by  M. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  P  =  (PPoints `  G )   &    |- 
 .~  =  ( (ss
 `  G ) `  M )   &    |-  L  =  (PLines `  G )   =>    |-  ( ph  ->  ( H  e.  ( (Halfplane `  G ) `  M ) 
 <-> 
 E. x  e.  ( P  \  M ) ( H  =  [ x ]  .~  \/  H  =  ( ( P  \  M )  \  [ x ]  .~  ) ) ) )
 
Theoremabhp1 26174 The half planes delimited by  M. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  P  =  (PPoints `  G )   &    |- 
 .~  =  ( (ss
 `  G ) `  M )   &    |-  L  =  (PLines `  G )   &    |-  ( ph  ->  A  e.  ( P  \  M ) )   =>    |-  ( ph  ->  ( H  e.  ( (Halfplane `  G ) `  M ) 
 <->  ( H  =  [ A ]  .~  \/  H  =  ( ( P  \  M )  \  [ A ]  .~  ) ) ) )
 
Theoremabhp2 26175 The half planes delimited by  M. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.) NEW
 |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  P  =  (PPoints `  G )   &    |- 
 .~  =  ( (ss
 `  G ) `  M )   &    |-  L  =  (PLines `  G )   &    |-  ( ph  ->  A  e.  ( P  \  M ) )   =>    |-  ( ph  ->  ( (Halfplane `  G ) `  M )  =  { [ A ]  .~  ,  ( ( P  \  M )  \  [ A ]  .~  ) } )
 
Theorembhp2a 26176* The half planes delimited by  M. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  P  =  (PPoints `  G )   &    |- 
 .~  =  ( (ss
 `  G ) `  M )   &    |-  L  =  (PLines `  G )   =>    |-  ( ph  ->  (
 (Halfplane `  G ) `  M )  =  { h  |  E. x  e.  ( P  \  M ) ( h  =  [ x ]  .~  \/  h  =  (
 ( P  \  M )  \  [ x ]  .~  ) ) } )
 
Theorembhp3 26177 Every line divides the plane into exactly two half planes . (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  P  =  (PPoints `  G )   &    |- 
 .~  =  ( (ss
 `  G ) `  M )   &    |-  L  =  (PLines `  G )   =>    |-  ( ph  ->  ( # `
  ( (Halfplane `  G ) `  M ) )  =  2 )
 
Syntaxcconvex 26178 Extend class notation with the convex symbol.
 class convex
 
Definitiondf-cnvx 26179* Definition of the convex subsets of points. Definition 24 of [AitkenIBG] p. 15. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
 |- convex  =  ( f  e. Ibg  |->  { x  e.  ~P (PPoints `  f
 )  |  A. p  e.  x  A. q  e.  x  ( p ( seg `  f )
 q )  C_  x } )
 
Syntaxcsegc 26180 Extend class notation with the segment congruence selector.
 class segc
 
Syntaxcibcg 26181 Extend class notation with the class of all planar, incidence, betweenness, congruence geometries.
 class Ibcg
 
Definitiondf-segc 26182 Segment congruence selector. (Contributed by FL, 1-Apr-2016.)
 |- segc  =  .r
 
Syntaxcangc 26183 Extend class notation with the angle congruence selector.
 class angc
 
Definitiondf-angc 26184 Angle congruence selector. (Contributed by FL, 1-Apr-2016.)
 |- angc  =  * r
 
Syntaxcangtrg 26185 Extend class notation a function returning an angle in a triangle.
 class angtrg
 
Definitiondf-angtrg 26186* Select an angle in a triangle. Definition 2 of [AitkenIBCG] p. 3. (Contributed by FL, 2-Sep-2016.)
 |- angtrg  =  ( g  e. Ig  |->  ( u  e.  ( (PPoints `  g
 )dWords 3 ) ,  x  e.  ( 1
 ... 3 )  |->  ( (angle `  g ) `  if ( x  =  2 ,  <" ( u `  1 ) ( u `  2 ) ( u `  3
 ) "> ,  if ( x  =  3 ,  <" ( u `
  2 ) ( u `  3 ) ( u `  1
 ) "> ,  <" ( u `  3
 ) ( u `  1 ) ( u `
  2 ) "> ) ) ) ) )
 
Syntaxcsegtrg 26187 Extend class notation a function returning a segment in a triangle.
 class segtrg
 
Definitiondf-segtrg 26188* Select a segment in a triangle. (Contributed by FL, 2-Sep-2016.)
 |- segtrg  =  ( g  e. Ig  |->  ( u  e.  ( (PPoints `  g
 )dWords 3 ) ,  x  e.  ( 1
 ... 3 )  |->  ( (angle `  g ) `  if ( x  =  1 ,  ( ( u `  1 ) ( seg `  g
 ) ( u `  2 ) ) ,  if ( x  =  2 ,  ( ( u `  2 ) ( seg `  g
 ) ( u `  3 ) ) ,  ( ( u `  3 ) ( seg `  g ) ( u `
  1 ) ) ) ) ) ) )
 
Definitiondf-trcng 26189* Congruence of two triangles. Triangles are congruent if their sides and angles are congruent. "Technically, congruence is not a property of triangles themselves, but of triangles with a given ordering of their vertices. Triangles that are congruent under one ordering, might not be under other orderings." Definition 2 of [AitkenIBCG] p. 3. (Contributed by FL, 2-Sep-2016.)
 |- trcng  =  ( g  e.  _V  |->  {
 <. u ,  v >.  |  ( ( u  e.  ( (PPoints `  g
 )dWords 3 )  /\  v  e.  ( (PPoints `  g )dWords 3 ) )  /\  ( ( u (segtrg `  g
 ) 1 ) (segc `  g ) ( v (segtrg `  g )
 1 )  /\  ( u (segtrg `  g )
 2 ) (segc `  g ) ( v (segtrg `  g )
 2 )  /\  ( u (segtrg `  g )
 3 ) (segc `  g ) ( v (segtrg `  g )
 3 ) )  /\  ( ( u (angtrg `  g ) 2 ) (angc `  g )
 ( v (angtrg `  g
 ) 2 )  /\  ( u (angtrg `  g
 ) 3 ) (angc `  g ) ( v (angtrg `  g )
 3 )  /\  ( u (angtrg `  g )
 1 ) (angc `  g ) ( v (angtrg `  g )
 1 ) ) ) } )
 
Definitiondf-ibcg 26190* Incidence-Betweenness Geometry plus congruence axioms. (Here is an excerpt of Aitken's handout.)

Axiom (C-1). Segment congruence is an equivalence relation for line segments.

Axiom (C-2). Suppose that 
A B is a line segment and  C D is a ray. Then there is a unique point  E on  C D, distinct from  C, such that  A B  .~  s C E. The next axiom concerns copying dividing or intermediate points on a segment.

Axiom (C-3). Suppose that 
A C and  A' C' are congruent line segments. If B is a point such that  A * B * C, then there is a point  B' such that  A' * B' * C',  A B  .~  s A' B', and  B C  .~  s B' C'.

Axiom (C-4). Angle congruence is an equivalence relation for angles.

Axiom (C-5). (Copying an angle) Suppose  B A C is an angle, and  D E is a ray. Then on any given side of  D E, there is a unique ray  D F such that  B A C  .~  a E D F.

Axiom (C-6). (Copying a triangle) Suppose  A B C is a triangle, and  A' B' is a segment such that  A B  .~  s A' B'. Then on any given side of  A' B', there is a point  C' such that  A B C  .~  a A' B' C'.

(For my private use only. Don't use.) Axiom C-1 C-2, C-3, C-4, C-5, C-6 of [AitkenIBCG] p. 2 . (Contributed by FL, 1-Apr-2016.)

 |- Ibcg  =  {
 g  e. Ibg  |  [. (angc `  g )  /  o ]. [. (PPoints `  g
 )  /  p ]. [. (btw `  g )  /  q ]. [. (ray `  g
 )  /  r ]. [. (segc `  g )  /  s ]. [. ( seg `  g )  /  t ]. [. ( line `  g )  /  l ]. [. (angle `  g
 )  /  u ]. [. (triangle `  g )  /  v ]. [. ( pdWords 3
 )  /  w ]. [. (Halfplane `  g )  /  x ].
 [. (Segments `  g )  /  y ]. [. (trcng `  g )  /  z ]. ( s  Er  y  /\  o  Er  ( u " w )  /\  A. d  e.  p  A. e  e.  p  (
 d  =/=  e  ->  (
 A. a  e.  p  A. b  e.  p  ( ( a  =/=  b  ->  E! f  e.  (
 d r e ) ( f  =/=  d  /\  ( a t b ) s ( d t f ) ) )  /\  A. c  e.  p  ( (
 c  e.  ( a q b )  /\  ( a t b ) s ( d t e ) ) 
 ->  E. f  e.  p  ( f  e.  (
 d q e ) 
 /\  ( a t c ) s ( d t f ) 
 /\  ( c t b ) s ( f t e ) ) ) )  /\  A. a  e.  w  (
 A. b  e.  ( x `  ( d l e ) ) E! f  e.  b  ( u `  a ) o ( u `  <" e d f "> )  /\  ( ( ( a `
  1 ) t ( a `  2
 ) ) s ( d t e ) 
 ->  A. b  e.  ( x `  ( d l e ) ) E. f  e.  b  a
 z ( v `  <" d e f "> ) ) ) ) ) ) }
 
Theoremisibcg 26191* The predicate "is a incidence betwenness geometry with congruences". (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  .~  a  =  (angc `  G )   &    |-  P  =  (PPoints `  G )   &    |-  U  =  (Segments `  G )   &    |-  B  =  (btw `  G )   &    |-  R  =  (ray `  G )   &    |-  .~  s  =  (segc `  G )   &    |-  S  =  ( seg `  G )   &    |-  L  =  ( line `  G )   &    |-  H  =  (Halfplane `  G )   &    |-  T  =  (triangle `  G )   &    |-  A  =  (angle `  G )   &    |-  .~  t  =  (trcng `  G )   &    |-  W  =  ( PdWords 3 )   =>    |-  ( G  e. Ibcg  <->  ( G  e. Ibg  /\  (  .~  s  Er  U  /\  .~  a  Er  ( A " W ) )  /\  A. d  e.  P  A. e  e.  P  ( d  =/=  e  ->  ( A. a  e.  P  A. b  e.  P  ( ( a  =/=  b  ->  E! f  e.  ( d R e ) ( f  =/=  d  /\  ( a S b )  .~  s ( d S f ) ) )  /\  A. c  e.  P  (
 ( c  e.  (
 a B b ) 
 /\  ( a S b )  .~  s
 ( d S e ) )  ->  E. f  e.  P  ( f  e.  ( d B e )  /\  ( a S c )  .~  s ( d S f )  /\  (
 c S b ) 
 .~  s ( f S e ) ) ) )  /\  A. a  e.  W  ( A. b  e.  ( H `  ( d L e ) ) E! f  e.  b  ( A `  a ) 
 .~  a ( A `
  <" e d f "> )  /\  ( ( ( a `
  1 ) S ( a `  2
 ) )  .~  s
 ( d S e )  ->  A. b  e.  ( H `  (
 d L e ) ) E. f  e.  b  a  .~  t
 ( T `  <" d
 e f "> ) ) ) ) ) ) )
 
Syntaxcslices 26192 Extend class notation with the slices symbol.
 class slices
 
Definitiondf-slices 26193* Return the slices generated by the Dedekind cut of a set of points. Definition 1 of [AitkenNG] p. 2. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
 |- slices  =  ( f  e. Ibg  |->  ( x  e.  ~P (PPoints `  f
 )  |->  { <. a ,  b >.  |  ( ( ( a  u.  b )  =  x  /\  (
 a  i^i  b )  =  (/) )  /\  (
 a  =/=  (/)  /\  b  =/= 
 (/) )  /\  (
 a  e.  (convex `  f
 )  /\  b  e.  (convex `  f ) ) ) } ) )
 
Syntaxccut 26194 Extend class notation with the cutpoint symbol.
 class Cut
 
Definitiondf-Cut 26195* Return the cutpoints of a set of points  ( x  u.  y
) where  x and  y are the slices of a Dedekind cut of  ( x  u.  y
). Definition 2 of [AitkenNG] p. 2. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
 |- Cut  =  ( f  e. Ibg  |->  ( x  e.  ~P (PPoints `  f
 ) ,  y  e. 
 ~P (PPoints `  f )  |->  { c  e.  ( x  u.  y )  | 
 A. u  e.  ( x  u.  y ) A. v  e.  ( x  u.  y ) ( u  e.  ( c (btw `  f ) v ) 
 ->  ( ( u  e.  x  /\  v  e.  x )  \/  ( u  e.  y  /\  v  e.  y )
 ) ) } )
 )
 
Syntaxcneug 26196 Extend class notation with the neutral geometry symbol.
 class Neug
 
Definitiondf-neug 26197* Definition of a neutral geometry. Every Dedekind cut of a line has a cut point. (Axiom of Dedekind in [AitkenNG] p. 3.) (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
 |- Neug  =  {
 g  e. Ibcg  |  A. l  e.  (PLines `  g ) A. u  e.  (
 (slices `  g ) `  l ) ( ( 1st `  u )
 (Cut `  g )
 ( 2nd `  u )
 )  =/=  (/) }
 
Syntaxccircle 26198 Extend class notation with the Circle symbol.
 class Circle
 
Definitiondf-crcl 26199* Definition of a circle (degenerated or not). Definition 5 of [AitkenNG] p. 4. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
 |- Circle  =  ( g  e. Ibcg  |->  ( x  e.  (PPoints `  g
 ) ,  y  e.  (PPoints `  g )  |->  { u  e.  (PPoints `  g )  |  ( x ( seg `  g
 ) u ) (segc `  g ) ( x ( seg `  g
 ) y ) }
 ) )
 
18.14  Mathbox for Jeff Hankins
 
18.14.1  Miscellany
 
Theorema1i13 26200 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ps  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
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