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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcolcom 24601 Swapping the points defining a line keeps it unchanged. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )   =>    |-  ( ph  ->  ( Z  e.  ( Y L X )  \/  Y  =  X ) )
 
Theoremcolrot1 24602 Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )   =>    |-  ( ph  ->  ( X  e.  ( Y L Z )  \/  Y  =  Z ) )
 
Theoremcolrot2 24603 Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )   =>    |-  ( ph  ->  ( Y  e.  ( Z L X )  \/  Z  =  X ) )
 
Theoremncolcom 24604 Swapping non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  -.  ( Z  e.  ( X L Y )  \/  X  =  Y ) )   =>    |-  ( ph  ->  -.  ( Z  e.  ( Y L X )  \/  Y  =  X ) )
 
Theoremncolrot1 24605 Rotating non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  -.  ( Z  e.  ( X L Y )  \/  X  =  Y ) )   =>    |-  ( ph  ->  -.  ( X  e.  ( Y L Z )  \/  Y  =  Z ) )
 
Theoremncolrot2 24606 Rotating non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  -.  ( Z  e.  ( X L Y )  \/  X  =  Y ) )   =>    |-  ( ph  ->  -.  ( Y  e.  ( Z L X )  \/  Z  =  X ) )
 
Theoremtgdim01ln 24607 In geometries of dimension lower than 2, any 3 points are colinear. (Contributed by Thierry Arnoux, 27-Aug-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  -.  GDimTarskiG 2 )   =>    |-  ( ph  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )
 
Theoremncoltgdim2 24608 If there are 3 non-colinear points, dimension must be 2 or more. tglowdim2l 24693 converse (Contributed by Thierry Arnoux, 23-Feb-2020.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  -.  ( Z  e.  ( X L Y )  \/  X  =  Y ) )   =>    |-  ( ph  ->  GDimTarskiG 2 )
 
Theoremlnxfr 24609 Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  ( Y  e.  ( X L Z )  \/  X  =  Z ) )   &    |-  ( ph  ->  <" X Y Z ">  .~  <" A B C "> )   =>    |-  ( ph  ->  ( B  e.  ( A L C )  \/  A  =  C ) )
 
Theoremlnext 24610* Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  ( Y  e.  ( X L Z )  \/  X  =  Z ) )   &    |-  ( ph  ->  ( X  .-  Y )  =  ( A  .-  B ) )   =>    |-  ( ph  ->  E. c  e.  P  <" X Y Z ">  .~  <" A B c "> )
 
Theoremtgfscgr 24611 Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  T  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  ( Y  e.  ( X L Z )  \/  X  =  Z ) )   &    |-  ( ph  ->  <" X Y Z ">  .~  <" A B C "> )   &    |-  ( ph  ->  ( X  .-  T )  =  ( A  .-  D ) )   &    |-  ( ph  ->  ( Y  .-  T )  =  ( B  .-  D )
 )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  ( Z  .-  T )  =  ( C  .-  D ) )
 
Theoremlncgr 24612 Congruence rule for lines. Theorem 4.17 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 28-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  ( Y  e.  ( X L Z )  \/  X  =  Z ) )   &    |-  ( ph  ->  ( X  .-  A )  =  ( X  .-  B ) )   &    |-  ( ph  ->  ( Y  .-  A )  =  ( Y  .-  B ) )   =>    |-  ( ph  ->  ( Z  .-  A )  =  ( Z  .-  B ) )
 
Theoremlnid 24613 Identity law for points on lines. Theorem 4.18 of [Schwabhauser] p. 38. (Contributed by Thierry Arnoux, 28-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  ( Y  e.  ( X L Z )  \/  X  =  Z ) )   &    |-  ( ph  ->  ( X  .-  Z )  =  ( X  .-  A ) )   &    |-  ( ph  ->  ( Y  .-  Z )  =  ( Y  .-  A ) )   =>    |-  ( ph  ->  Z  =  A )
 
Theoremtgidinside 24614 Law for finding a point inside a segment. Theorem 4.19 of [Schwabhauser] p. 38. (Contributed by Thierry Arnoux, 28-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  Z  e.  ( X I Y ) )   &    |-  ( ph  ->  ( X  .-  Z )  =  ( X  .-  A ) )   &    |-  ( ph  ->  ( Y  .-  Z )  =  ( Y  .-  A )
 )   =>    |-  ( ph  ->  Z  =  A )
 
15.2.8  Connectivity of betweenness
 
Theoremtgbtwnconn1lem1 24615 Lemma for tgbtwnconn1 24618 (Contributed by Thierry Arnoux, 30-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  B  e.  ( A I D ) )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  E  e.  P )   &    |-  ( ph  ->  F  e.  P )   &    |-  ( ph  ->  H  e.  P )   &    |-  ( ph  ->  J  e.  P )   &    |-  ( ph  ->  D  e.  ( A I E ) )   &    |-  ( ph  ->  C  e.  ( A I F ) )   &    |-  ( ph  ->  E  e.  ( A I H ) )   &    |-  ( ph  ->  F  e.  ( A I J ) )   &    |-  ( ph  ->  ( E  .-  D )  =  ( C  .-  D ) )   &    |-  ( ph  ->  ( C  .-  F )  =  ( C  .-  D ) )   &    |-  ( ph  ->  ( E  .-  H )  =  ( B  .-  C ) )   &    |-  ( ph  ->  ( F  .-  J )  =  ( B  .-  D ) )   =>    |-  ( ph  ->  H  =  J )
 
Theoremtgbtwnconn1lem2 24616 Lemma for tgbtwnconn1 24618 (Contributed by Thierry Arnoux, 30-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  B  e.  ( A I D ) )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  E  e.  P )   &    |-  ( ph  ->  F  e.  P )   &    |-  ( ph  ->  H  e.  P )   &    |-  ( ph  ->  J  e.  P )   &    |-  ( ph  ->  D  e.  ( A I E ) )   &    |-  ( ph  ->  C  e.  ( A I F ) )   &    |-  ( ph  ->  E  e.  ( A I H ) )   &    |-  ( ph  ->  F  e.  ( A I J ) )   &    |-  ( ph  ->  ( E  .-  D )  =  ( C  .-  D ) )   &    |-  ( ph  ->  ( C  .-  F )  =  ( C  .-  D ) )   &    |-  ( ph  ->  ( E  .-  H )  =  ( B  .-  C ) )   &    |-  ( ph  ->  ( F  .-  J )  =  ( B  .-  D ) )   =>    |-  ( ph  ->  ( E  .-  F )  =  ( C  .-  D ) )
 
Theoremtgbtwnconn1lem3 24617 Lemma for tgbtwnconn1 24618 (Contributed by Thierry Arnoux, 30-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  B  e.  ( A I D ) )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  E  e.  P )   &    |-  ( ph  ->  F  e.  P )   &    |-  ( ph  ->  H  e.  P )   &    |-  ( ph  ->  J  e.  P )   &    |-  ( ph  ->  D  e.  ( A I E ) )   &    |-  ( ph  ->  C  e.  ( A I F ) )   &    |-  ( ph  ->  E  e.  ( A I H ) )   &    |-  ( ph  ->  F  e.  ( A I J ) )   &    |-  ( ph  ->  ( E  .-  D )  =  ( C  .-  D ) )   &    |-  ( ph  ->  ( C  .-  F )  =  ( C  .-  D ) )   &    |-  ( ph  ->  ( E  .-  H )  =  ( B  .-  C ) )   &    |-  ( ph  ->  ( F  .-  J )  =  ( B  .-  D ) )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  X  e.  ( C I E ) )   &    |-  ( ph  ->  X  e.  ( D I F ) )   &    |-  ( ph  ->  C  =/=  E )   =>    |-  ( ph  ->  D  =  F )
 
Theoremtgbtwnconn1 24618 Connectivity law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. In earlier presentations of Tarski's axioms, this theorem appeared as an additional axiom. It was derived from the other axioms by Gupta, 1965. (Contributed by Thierry Arnoux, 30-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  B  e.  ( A I D ) )   =>    |-  ( ph  ->  ( C  e.  ( A I D )  \/  D  e.  ( A I C ) ) )
 
Theoremtgbtwnconn2 24619 Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 17-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  B  e.  ( A I D ) )   =>    |-  ( ph  ->  ( C  e.  ( B I D )  \/  D  e.  ( B I C ) ) )
 
Theoremtgbtwnconn3 24620 Inner connectivity law for betweenness. Theorem 5.3 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 17-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  B  e.  ( A I D ) )   &    |-  ( ph  ->  C  e.  ( A I D ) )   =>    |-  ( ph  ->  ( B  e.  ( A I C )  \/  C  e.  ( A I B ) ) )
 
Theoremtgbtwnconnln3 24621 Derive colinearity from betweenness (Contributed by Thierry Arnoux, 17-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  B  e.  ( A I D ) )   &    |-  ( ph  ->  C  e.  ( A I D ) )   &    |-  L  =  (LineG `  G )   =>    |-  ( ph  ->  ( B  e.  ( A L C )  \/  A  =  C ) )
 
Theoremtgbtwnconn22 24622 Double connectivity law for betweenness (Contributed by Thierry Arnoux, 1-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  E  e.  P )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  C  =/=  B )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  B  e.  ( A I D ) )   &    |-  ( ph  ->  B  e.  ( C I E ) )   =>    |-  ( ph  ->  B  e.  ( D I E ) )
 
Theoremtgbtwnconnln1 24623 Derive colinearity from betweenness (Contributed by Thierry Arnoux, 17-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  B  e.  ( A I D ) )   =>    |-  ( ph  ->  ( A  e.  ( C L D )  \/  C  =  D ) )
 
Theoremtgbtwnconnln2 24624 Derive colinearity from betweenness (Contributed by Thierry Arnoux, 17-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  B  e.  ( A I D ) )   =>    |-  ( ph  ->  ( B  e.  ( C L D )  \/  C  =  D ) )
 
15.2.9  Less-than relation in geometric congruences
 
Syntaxcleg 24625 Less-than relation for geometric congruences.
 class ≤G
 
Definitiondf-leg 24626* Define the less-than relationship between geometric distance congruence classes. See legval 24627. (Contributed by Thierry Arnoux, 21-Jun-2019.)
 |- ≤G  =  ( g  e.  _V  |->  {
 <. e ,  f >.  | 
 [. ( Base `  g
 )  /  p ]. [. ( dist `  g )  /  d ]. [. (Itv `  g )  /  i ]. E. x  e.  p  E. y  e.  p  ( f  =  ( x d y ) 
 /\  E. z  e.  p  ( z  e.  ( x i y ) 
 /\  e  =  ( x d z ) ) ) } )
 
Theoremlegval 24627* Value of the less-than relationship. (Contributed by Thierry Arnoux, 21-Jun-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   =>    |-  ( ph  ->  .<_  =  { <. e ,  f >.  |  E. x  e.  P  E. y  e.  P  ( f  =  ( x  .-  y
 )  /\  E. z  e.  P  ( z  e.  ( x I y )  /\  e  =  ( x  .-  z
 ) ) ) }
 )
 
Theoremlegov 24628* Value of the less-than relationship. Definition 5.4 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 21-Jun-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   =>    |-  ( ph  ->  (
 ( A  .-  B )  .<_  ( C  .-  D )  <->  E. z  e.  P  ( z  e.  ( C I D )  /\  ( A  .-  B )  =  ( C  .-  z ) ) ) )
 
Theoremlegov2 24629* An equivalent definition of the less-than relationship. Definition 5.5 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 21-Jun-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   =>    |-  ( ph  ->  (
 ( A  .-  B )  .<_  ( C  .-  D )  <->  E. x  e.  P  ( B  e.  ( A I x )  /\  ( A  .-  x )  =  ( C  .-  D ) ) ) )
 
Theoremlegid 24630 Reflexivity of the less-than relationship. Proposition 5.7 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( A  .-  B )  .<_  ( A  .-  B )
 )
 
Theorembtwnleg 24631 Betweenness implies less-than relation. (Contributed by Thierry Arnoux, 3-Jul-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  B  e.  ( A I C ) )   =>    |-  ( ph  ->  ( A  .-  B )  .<_  ( A 
 .-  C ) )
 
Theoremlegtrd 24632 Transitivity of the less-than relationship. Proposition 5.8 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  E  e.  P )   &    |-  ( ph  ->  F  e.  P )   &    |-  ( ph  ->  ( A  .-  B )  .<_  ( C 
 .-  D ) )   &    |-  ( ph  ->  ( C  .-  D )  .<_  ( E 
 .-  F ) )   =>    |-  ( ph  ->  ( A  .-  B )  .<_  ( E 
 .-  F ) )
 
Theoremlegtri3 24633 Equality from the less-than relationship. Proposition 5.9 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  ( A  .-  B )  .<_  ( C  .-  D )
 )   &    |-  ( ph  ->  ( C  .-  D )  .<_  ( A  .-  B )
 )   =>    |-  ( ph  ->  ( A  .-  B )  =  ( C  .-  D ) )
 
Theoremlegtrid 24634 Trichotomy law for the less-than relationship. Proposition 5.10 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   =>    |-  ( ph  ->  (
 ( A  .-  B )  .<_  ( C  .-  D )  \/  ( C  .-  D )  .<_  ( A  .-  B )
 ) )
 
Theoremleg0 24635 Degenerated (zero-length) segments are minimal. Proposition 5.11 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   =>    |-  ( ph  ->  ( A  .-  A )  .<_  ( C  .-  D )
 )
 
Theoremlegeq 24636 Deduce equality from "less than" null segments (Contributed by Thierry Arnoux, 12-Aug-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  ( A  .-  B )  .<_  ( C  .-  C )
 )   =>    |-  ( ph  ->  A  =  B )
 
Theoremlegbtwn 24637 Deduce betweenness from "less than" relation. Corresponds loosely to Proposition 6.13 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 25-Aug-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) )   &    |-  ( ph  ->  ( C  .-  A )  .<_  ( C 
 .-  B ) )   =>    |-  ( ph  ->  A  e.  ( C I B ) )
 
Theoremtgcgrsub2 24638 Removing identical parts from the end of a line segment preserves congruence. In this version the order of points is not known. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  E  e.  P )   &    |-  ( ph  ->  F  e.  P )   &    |-  ( ph  ->  ( B  e.  ( A I C )  \/  C  e.  ( A I B ) ) )   &    |-  ( ph  ->  ( E  e.  ( D I F )  \/  F  e.  ( D I E ) ) )   &    |-  ( ph  ->  ( A  .-  B )  =  ( D  .-  E ) )   &    |-  ( ph  ->  ( A  .-  C )  =  ( D  .-  F ) )   =>    |-  ( ph  ->  ( B  .-  C )  =  ( E  .-  F ) )
 
Theoremltgseg 24639* The set  E denotes the possible values of the congruence. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  E  =  (  .-  " ( P  X.  P ) )   &    |-  ( ph  ->  Fun  .-  )   &    |-  ( ph  ->  A  e.  E )   =>    |-  ( ph  ->  E. x  e.  P  E. y  e.  P  A  =  ( x  .-  y )
 )
 
Theoremltgov 24640 Strict "shorter than" geometric relation between segments. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  E  =  (  .-  " ( P  X.  P ) )   &    |-  ( ph  ->  Fun  .-  )   &    |-  .<  =  ( (  .<_  |`  E ) 
 \  _I  )   &    |-  ( ph  ->  ( P  X.  P )  C_  dom  .-  )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  (
 ( A  .-  B )  .<  ( C  .-  D )  <->  ( ( A 
 .-  B )  .<_  ( C  .-  D )  /\  ( A  .-  B )  =/=  ( C  .-  D ) ) ) )
 
Theoremlegov3 24641 An equivalent definition of the less-than relationship, from the strict relation. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  E  =  (  .-  " ( P  X.  P ) )   &    |-  ( ph  ->  Fun  .-  )   &    |-  .<  =  ( (  .<_  |`  E ) 
 \  _I  )   &    |-  ( ph  ->  ( P  X.  P )  C_  dom  .-  )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  (
 ( A  .-  B )  .<_  ( C  .-  D )  <->  ( ( A 
 .-  B )  .<  ( C  .-  D )  \/  ( A  .-  B )  =  ( C  .-  D ) ) ) )
 
Theoremlegso 24642 The shorter-than relationship builds an order over pairs. Remark 5.13 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .<_  =  (≤G `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  E  =  (  .-  " ( P  X.  P ) )   &    |-  ( ph  ->  Fun  .-  )   &    |-  .<  =  ( (  .<_  |`  E ) 
 \  _I  )   &    |-  ( ph  ->  ( P  X.  P )  C_  dom  .-  )   =>    |-  ( ph  ->  .<  Or  E )
 
15.2.10  Rays
 
Syntaxchlg 24643 Function producing the relation "belong to the same half-line".
 class hlG
 
Definitiondf-hlg 24644* Define the function producting the relation "belong to the same half-line" (Contributed by Thierry Arnoux, 15-Aug-2020.)
 |- hlG 
 =  ( g  e. 
 _V  |->  ( c  e.  ( Base `  g )  |->  { <. a ,  b >.  |  ( ( a  e.  ( Base `  g
 )  /\  b  e.  ( Base `  g )
 )  /\  ( a  =/=  c  /\  b  =/=  c  /\  ( a  e.  ( c (Itv `  g ) b )  \/  b  e.  (
 c (Itv `  g
 ) a ) ) ) ) } )
 )
 
Theoremishlg 24645 Rays : Definition 6.1 of [Schwabhauser] p. 43. With this definition,  A ( K `
 C ) B means that  A and  B are on the same ray with initial point  C. This follows the same notation as Schwabhauser where rays are first defined as a relation. It is possible to recover the ray itself using e.g.  ( ( K `  C ) " { A } ) (Contributed by Thierry Arnoux, 21-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e.  V )   =>    |-  ( ph  ->  ( A ( K `  C ) B  <->  ( A  =/=  C 
 /\  B  =/=  C  /\  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) ) ) )
 
Theoremhlcomb 24646 The half-line relation commutes. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e.  V )   =>    |-  ( ph  ->  ( A ( K `  C ) B  <->  B ( K `  C ) A ) )
 
Theoremhlcomd 24647 The half-line relation commutes. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  A ( K `
  C ) B )   =>    |-  ( ph  ->  B ( K `  C ) A )
 
Theoremhlne1 24648 The half-line relation implies inequality. (Contributed by Thierry Arnoux, 22-Feb-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  A ( K `
  C ) B )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremhlne2 24649 The half-line relation implies inequality. (Contributed by Thierry Arnoux, 22-Feb-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  A ( K `
  C ) B )   =>    |-  ( ph  ->  B  =/=  C )
 
Theoremhlln 24650 The half-line relation implies colinearity, part of Theorem 6.4 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 22-Feb-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  A ( K `  C ) B )   =>    |-  ( ph  ->  A  e.  ( B L C ) )
 
Theoremhleqnid 24651 The endpoint does not belong to the half-line. (Contributed by Thierry Arnoux, 3-Mar-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e. TarskiG )   =>    |-  ( ph  ->  -.  A ( K `  A ) B )
 
Theoremhlid 24652 The half-line relation is reflexive. Theorem 6.5 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  A ( K `  C ) A )
 
Theoremhltr 24653 The half-line relation is transitive. Theorem 6.7 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 23-Feb-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  A ( K `
  D ) B )   &    |-  ( ph  ->  B ( K `  D ) C )   =>    |-  ( ph  ->  A ( K `  D ) C )
 
Theoremhlbtwn 24654 Betweenness is a sufficient condition to swap half-lines. (Contributed by Thierry Arnoux, 21-Feb-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  D  e.  ( C I B ) )   &    |-  ( ph  ->  B  =/=  C )   &    |-  ( ph  ->  D  =/=  C )   =>    |-  ( ph  ->  ( A ( K `  C ) B  <->  A ( K `  C ) D ) )
 
Theorembtwnhl1 24655 Deduce half-line from betweenness. (Contributed by Thierry Arnoux, 4-Mar-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  C  e.  ( A I B ) )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  C  =/=  A )   =>    |-  ( ph  ->  C ( K `  A ) B )
 
Theorembtwnhl2 24656 Deduce half-line from betweenness. (Contributed by Thierry Arnoux, 4-Mar-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  C  e.  ( A I B ) )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  C  =/=  B )   =>    |-  ( ph  ->  C ( K `  B ) A )
 
Theorembtwnhl 24657 Swap betweenness for a half-line. (Contributed by Thierry Arnoux, 2-Mar-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  A ( K `
  D ) B )   &    |-  ( ph  ->  D  e.  ( A I C ) )   =>    |-  ( ph  ->  D  e.  ( B I C ) )
 
Theoremlnhl 24658 Either a point  C on the line AB is on the same side as  A or on the opposite side. (Contributed by Thierry Arnoux, 21-Sep-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  D  e.  P )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  C  e.  ( A L B ) )   =>    |-  ( ph  ->  ( C ( K `  B ) A  \/  B  e.  ( A I C ) ) )
 
Theoremhlcgrex 24659* Construct a point on a half-line, at a given distance of its origin. (Contributed by Thierry Arnoux, 1-Aug-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  D  e.  P )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  D  =/=  A )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  E. x  e.  P  ( x ( K `  A ) D  /\  ( A 
 .-  x )  =  ( B  .-  C ) ) )
 
Theoremhlcgreulem 24660 Lemma for hlcgreu 24661 (Contributed by Thierry Arnoux, 9-Aug-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  D  e.  P )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  D  =/=  A )   &    |-  ( ph  ->  B  =/=  C )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X ( K `  A ) D )   &    |-  ( ph  ->  Y ( K `  A ) D )   &    |-  ( ph  ->  ( A  .-  X )  =  ( B  .-  C ) )   &    |-  ( ph  ->  ( A  .-  Y )  =  ( B  .-  C ) )   =>    |-  ( ph  ->  X  =  Y )
 
Theoremhlcgreu 24661* The point constructed in hlcgrex 24659 is unique. Theorem 6.11 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 9-Aug-2020.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  K  =  (hlG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  D  e.  P )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  D  =/=  A )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  E! x  e.  P  ( x ( K `  A ) D  /\  ( A  .-  x )  =  ( B  .-  C ) ) )
 
15.2.11  Lines
 
Theorembtwnlng1 24662 Betweenness implies colinearity (Contributed by Thierry Arnoux, 28-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Z  e.  ( X I Y ) )   =>    |-  ( ph  ->  Z  e.  ( X L Y ) )
 
Theorembtwnlng2 24663 Betweenness implies colinearity (Contributed by Thierry Arnoux, 28-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  X  e.  ( Z I Y ) )   =>    |-  ( ph  ->  Z  e.  ( X L Y ) )
 
Theorembtwnlng3 24664 Betweenness implies colinearity (Contributed by Thierry Arnoux, 28-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e.  ( X I Z ) )   =>    |-  ( ph  ->  Z  e.  ( X L Y ) )
 
Theoremlncom 24665 Swapping the points defining a line keeps it unchanged. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Z  e.  ( Y L X ) )   =>    |-  ( ph  ->  Z  e.  ( X L Y ) )
 
Theoremlnrot1 24666 Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e.  ( Z L X ) )   &    |-  ( ph  ->  Z  =/=  X )   =>    |-  ( ph  ->  Z  e.  ( X L Y ) )
 
Theoremlnrot2 24667 Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  X  e.  ( Y L Z ) )   &    |-  ( ph  ->  Y  =/=  Z )   =>    |-  ( ph  ->  Z  e.  ( X L Y ) )
 
Theoremncolne1 24668 Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  -.  ( X  e.  ( Y L Z )  \/  Y  =  Z ) )   =>    |-  ( ph  ->  X  =/=  Y )
 
Theoremncolne2 24669 Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 24669 could be simplified out and deleted, replaced by ncolcom 24604.
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  -.  ( X  e.  ( Y L Z )  \/  Y  =  Z ) )   =>    |-  ( ph  ->  X  =/=  Z )
 
Theoremtgisline 24670* The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  ran  L )   =>    |-  ( ph  ->  E. x  e.  B  E. y  e.  B  ( A  =  ( x L y )  /\  x  =/=  y ) )
 
Theoremtglnne 24671 It takes two different points to form a line. (Contributed by Thierry Arnoux, 27-Nov-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  ( X L Y )  e. 
 ran  L )   =>    |-  ( ph  ->  X  =/=  Y )
 
Theoremtglndim0 24672 There are no lines in dimension 0. (Contributed by Thierry Arnoux, 18-Oct-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  ( # `  B )  =  1 )   =>    |-  ( ph  ->  -.  A  e.  ran  L )
 
Theoremtgelrnln 24673 The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  ( X L Y )  e. 
 ran  L )
 
Theoremtglineeltr 24674 Transitivity law for lines, one half of tglineelsb2 24675 (Contributed by Thierry Arnoux, 25-May-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  Q  e.  B )   &    |-  ( ph  ->  P  =/=  Q )   &    |-  ( ph  ->  S  e.  B )   &    |-  ( ph  ->  S  =/=  P )   &    |-  ( ph  ->  S  e.  ( P L Q ) )   &    |-  ( ph  ->  R  e.  B )   &    |-  ( ph  ->  R  e.  ( P L S ) )   =>    |-  ( ph  ->  R  e.  ( P L Q ) )
 
Theoremtglineelsb2 24675 If  S lies on PQ , then PQ = PS . Theorem 6.16 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  Q  e.  B )   &    |-  ( ph  ->  P  =/=  Q )   &    |-  ( ph  ->  S  e.  B )   &    |-  ( ph  ->  S  =/=  P )   &    |-  ( ph  ->  S  e.  ( P L Q ) )   =>    |-  ( ph  ->  ( P L Q )  =  ( P L S ) )
 
Theoremtglinerflx1 24676 Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  Q  e.  B )   &    |-  ( ph  ->  P  =/=  Q )   =>    |-  ( ph  ->  P  e.  ( P L Q ) )
 
Theoremtglinerflx2 24677 Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  Q  e.  B )   &    |-  ( ph  ->  P  =/=  Q )   =>    |-  ( ph  ->  Q  e.  ( P L Q ) )
 
Theoremtglinecom 24678 Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  Q  e.  B )   &    |-  ( ph  ->  P  =/=  Q )   =>    |-  ( ph  ->  ( P L Q )  =  ( Q L P ) )
 
Theoremtglinethru 24679 If  A is a line containing two distinct points  P and  Q, then  A is the line through  P and  Q. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 25-May-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  Q  e.  B )   &    |-  ( ph  ->  P  =/=  Q )   &    |-  ( ph  ->  P  =/=  Q )   &    |-  ( ph  ->  A  e.  ran  L )   &    |-  ( ph  ->  P  e.  A )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  A  =  ( P L Q ) )
 
Theoremtghilberti1 24680* There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Thierry Arnoux, 25-May-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  Q  e.  B )   &    |-  ( ph  ->  P  =/=  Q )   =>    |-  ( ph  ->  E. x  e.  ran  L ( P  e.  x  /\  Q  e.  x ) )
 
Theoremtghilberti2 24681* There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Thierry Arnoux, 25-May-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  Q  e.  B )   &    |-  ( ph  ->  P  =/=  Q )   =>    |-  ( ph  ->  E* x  e.  ran  L ( P  e.  x  /\  Q  e.  x ) )
 
Theoremtglinethrueu 24682* There is a unique line going through any two distinct points. Theorem 6.19 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 25-May-2019.)
 |-  B  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  Q  e.  B )   &    |-  ( ph  ->  P  =/=  Q )   =>    |-  ( ph  ->  E! x  e.  ran  L ( P  e.  x  /\  Q  e.  x )
 )
 
Theoremtglnne0 24683 A line  A has at least one point. (Contributed by Thierry Arnoux, 4-Mar-2020.)
 |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  ran  L )   =>    |-  ( ph  ->  A  =/=  (/) )
 
Theoremtglnpt2 24684* Find a second point on a line. (Contributed by Thierry Arnoux, 18-Oct-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  ran  L )   &    |-  ( ph  ->  X  e.  A )   =>    |-  ( ph  ->  E. y  e.  A  X  =/=  y
 )
 
Theoremtglineintmo 24685* Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 25-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  ran  L )   &    |-  ( ph  ->  B  e.  ran  L )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  E* x ( x  e.  A  /\  x  e.  B ) )
 
Theoremtglineineq 24686 Two distinct lines intersect in at most one point, variation. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 6-Aug-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  ran  L )   &    |-  ( ph  ->  B  e.  ran  L )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  X  e.  ( A  i^i  B ) )   &    |-  ( ph  ->  Y  e.  ( A  i^i  B ) )   =>    |-  ( ph  ->  X  =  Y )
 
Theoremtglineneq 24687 Given three non-colinear points, build two different lines. (Contributed by Thierry Arnoux, 6-Aug-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  -.  ( A  e.  ( B L C )  \/  B  =  C ) )   =>    |-  ( ph  ->  ( A L B )  =/=  ( C L D ) )
 
Theoremtglineinteq 24688 Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 6-Aug-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  -.  ( A  e.  ( B L C )  \/  B  =  C ) )   &    |-  ( ph  ->  X  e.  ( A L B ) )   &    |-  ( ph  ->  Y  e.  ( A L B ) )   &    |-  ( ph  ->  X  e.  ( C L D ) )   &    |-  ( ph  ->  Y  e.  ( C L D ) )   =>    |-  ( ph  ->  X  =  Y )
 
Theoremncolncol 24689 Deduce non-colinearity from non-colinearity and colinearity. (Contributed by Thierry Arnoux, 27-Aug-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  -.  ( A  e.  ( B L C )  \/  B  =  C ) )   &    |-  ( ph  ->  D  e.  ( A L B ) )   &    |-  ( ph  ->  D  =/=  B )   =>    |-  ( ph  ->  -.  ( D  e.  ( B L C )  \/  B  =  C ) )
 
Theoremcoltr 24690 A transitivity law for colinearity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  A  e.  ( B L C ) )   &    |-  ( ph  ->  ( B  e.  ( C L D )  \/  C  =  D ) )   =>    |-  ( ph  ->  ( A  e.  ( C L D )  \/  C  =  D ) )
 
Theoremcoltr3 24691 A transitivity law for colinearity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  A  e.  ( B L C ) )   &    |-  ( ph  ->  D  e.  ( A I C ) )   =>    |-  ( ph  ->  D  e.  ( B L C ) )
 
Theoremcolline 24692* Three points are colinear iff there is a line through all three of them. Theorem 6.23 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 28-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  2 
 <_  ( # `  P ) )   =>    |-  ( ph  ->  (
 ( X  e.  ( Y L Z )  \/  Y  =  Z )  <->  E. a  e.  ran  L ( X  e.  a  /\  Y  e.  a  /\  Z  e.  a )
 ) )
 
Theoremtglowdim2l 24693* Reformulation of the lower dimension axiom for dimension 2. There exist three non colinear points. Theorem 6.24 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 30-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   =>    |-  ( ph  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  -.  ( c  e.  ( a L b )  \/  a  =  b ) )
 
Theoremtglowdim2ln 24694* There is always one point outside of any line. Theorem 6.25 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 16-Nov-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  E. c  e.  P  -.  c  e.  ( A L B ) )
 
15.2.12  Point inversions
 
Syntaxcmir 24695 Declare the constant for the point inversion function.
 class pInvG
 
Definitiondf-mir 24696* Define the point inversion ("mirror") function. Definition 7.5 of [Schwabhauser] p. 49. See mirval 24698 and ismir 24702. (Contributed by Thierry Arnoux, 30-May-2019.)
 |- pInvG  =  ( g  e.  _V  |->  ( m  e.  ( Base `  g )  |->  ( a  e.  ( Base `  g )  |->  ( iota_ b  e.  ( Base `  g
 ) ( ( m ( dist `  g )
 b )  =  ( m ( dist `  g
 ) a )  /\  m  e.  ( b
 (Itv `  g )
 a ) ) ) ) ) )
 
Theoremmirreu3 24697* Existential uniqueness of the mirror point. Theorem 7.8 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 30-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  M  e.  P )   =>    |-  ( ph  ->  E! b  e.  P  (
 ( M  .-  b
 )  =  ( M 
 .-  A )  /\  M  e.  ( b I A ) ) )
 
Theoremmirval 24698* Value of the point inversion function  S. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 30-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  S  =  (pInvG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   =>    |-  ( ph  ->  ( S `  A )  =  ( y  e.  P  |->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  y )  /\  A  e.  ( z I y ) ) ) ) )
 
Theoremmirfv 24699* Value of the point inversion function  M. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 30-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  S  =  (pInvG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  M  =  ( S `
  A )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( M `  B )  =  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) ) )
 
Theoremmircgr 24700 Property of the image by the point inversion function. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  S  =  (pInvG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  M  =  ( S `
  A )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( A  .-  ( M `  B ) )  =  ( A  .-  B ) )
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