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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtgcgrcomr 24601 Congruence commutes on the RHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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 )  =  ( C  .-  D ) )   =>    |-  ( ph  ->  ( A  .-  B )  =  ( D  .-  C ) )
 
Theoremtgcgrcoml 24602 Congruence commutes on the LHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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 )  =  ( C  .-  D ) )   =>    |-  ( ph  ->  ( B  .-  A )  =  ( C  .-  D ) )
 
Theoremtgcgrcomlr 24603 Congruence commutes on both sides. (Contributed by Thierry Arnoux, 23-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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 )  =  ( C  .-  D ) )   =>    |-  ( ph  ->  ( B  .-  A )  =  ( D  .-  C ) )
 
Theoremtgcgreqb 24604 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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 )  =  ( C  .-  D ) )   =>    |-  ( ph  ->  ( A  =  B  <->  C  =  D ) )
 
Theoremtgcgreq 24605 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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 )  =  ( C  .-  D ) )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  C  =  D )
 
Theoremtgcgrneq 24606 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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 )  =  ( C  .-  D ) )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  C  =/=  D )
 
Theoremtgcgrtriv 24607 Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( A  .-  A )  =  ( B  .-  B ) )
 
Theoremtgcgrextend 24608 Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) (Shortened by David A. Wheeler and Thierry Arnoux, 22-Apr-2020.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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  ->  F  e.  P )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  E  e.  ( D I F ) )   &    |-  ( ph  ->  ( A  .-  B )  =  ( D  .-  E ) )   &    |-  ( ph  ->  ( B  .-  C )  =  ( E  .-  F ) )   =>    |-  ( ph  ->  ( A  .-  C )  =  ( D  .-  F ) )
 
Theoremtgsegconeq 24609 Two points that satisfy the conclusion of axtgsegcon 24591 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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  ->  F  e.  P )   &    |-  ( ph  ->  D  =/=  A )   &    |-  ( ph  ->  A  e.  ( D I E ) )   &    |-  ( ph  ->  A  e.  ( D I F ) )   &    |-  ( ph  ->  ( A  .-  E )  =  ( B  .-  C ) )   &    |-  ( ph  ->  ( A  .-  F )  =  ( B  .-  C )
 )   =>    |-  ( ph  ->  E  =  F )
 
15.2.2  Betweenness
 
Theoremtgbtwntriv2 24610 Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  B  e.  ( A I B ) )
 
Theoremtgbtwncom 24611 Betweenness commutes. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  B  e.  ( A I C ) )   =>    |-  ( ph  ->  B  e.  ( C I A ) )
 
Theoremtgbtwncomb 24612 Betweenness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   =>    |-  ( ph  ->  ( B  e.  ( A I C )  <->  B  e.  ( C I A ) ) )
 
Theoremtgbtwnne 24613 Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  B  =/=  A )   =>    |-  ( ph  ->  A  =/=  C )
 
Theoremtgbtwntriv1 24614 Betweenness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  A  e.  ( A I B ) )
 
Theoremtgbtwnswapid 24615 If you can swap the first two arguments of a betweenness statement, then those arguments are identical. Theorem 3.4 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 16-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  A  e.  ( B I C ) )   &    |-  ( ph  ->  B  e.  ( A I C ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremtgbtwnintr 24616 Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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  e.  ( B I D ) )   &    |-  ( ph  ->  B  e.  ( C I D ) )   =>    |-  ( ph  ->  B  e.  ( A I C ) )
 
Theoremtgbtwnexch3 24617 Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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 C ) )   &    |-  ( ph  ->  C  e.  ( A I D ) )   =>    |-  ( ph  ->  C  e.  ( B I D ) )
 
Theoremtgbtwnouttr2 24618 Outer transitivity law for betweenness. Left-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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  =/=  C )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  C  e.  ( B I D ) )   =>    |-  ( ph  ->  C  e.  ( A I D ) )
 
Theoremtgbtwnexch2 24619 Exchange the outer point of two betweenness statements. Right-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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.  ( B I D ) )   =>    |-  ( ph  ->  C  e.  ( A I D ) )
 
Theoremtgbtwnouttr 24620 Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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  =/=  C )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  C  e.  ( B I D ) )   =>    |-  ( ph  ->  B  e.  ( A I D ) )
 
Theoremtgbtwnexch 24621 Outer transitivity law for betweenness. Right-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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 C ) )   &    |-  ( ph  ->  C  e.  ( A I D ) )   =>    |-  ( ph  ->  B  e.  ( A I D ) )
 
Theoremtgtrisegint 24622* A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Thierry Arnoux, 23-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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  ->  F  e.  P )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  E  e.  ( D I C ) )   &    |-  ( ph  ->  F  e.  ( A I D ) )   =>    |-  ( ph  ->  E. q  e.  P  ( q  e.  ( F I C )  /\  q  e.  ( B I E ) ) )
 
15.2.3  Dimension
 
Theoremtglowdim1 24623* Lower dimension axiom for one dimension. In dimension at least 1, there are at least two distinct points. The condition "the space is of dimension 1 or more" is written here as  2  <_  ( # `  P
) to avoid a new definition, but a different convention could be chosen. (Contributed by Thierry Arnoux, 23-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  2  <_  ( # `
  P ) )   =>    |-  ( ph  ->  E. x  e.  P  E. y  e.  P  x  =/=  y
 )
 
Theoremtglowdim1i 24624* Lower dimension axiom for one dimension. (Contributed by Thierry Arnoux, 28-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  2  <_  ( # `
  P ) )   &    |-  ( ph  ->  X  e.  P )   =>    |-  ( ph  ->  E. y  e.  P  X  =/=  y
 )
 
Theoremtgldimor 24625 Excluded-middle like statement allowing to treat dimension zero as a special case. (Contributed by Thierry Arnoux, 11-Apr-2019.)
 |-  P  =  ( E `
  F )   &    |-  ( ph  ->  A  e.  P )   =>    |-  ( ph  ->  (
 ( # `  P )  =  1  \/  2  <_  ( # `  P ) ) )
 
Theoremtgldim0eq 24626 In dimension zero, any two points are equal. (Contributed by Thierry Arnoux, 11-Apr-2019.)
 |-  P  =  ( E `
  F )   &    |-  ( ph  ->  ( # `  P )  =  1 )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  A  =  B )
 
Theoremtgldim0itv 24627 In dimension zero, any two points are equal. (Contributed by Thierry Arnoux, 12-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  ( # `  P )  =  1 )   =>    |-  ( ph  ->  A  e.  ( B I C ) )
 
Theoremtgldim0cgr 24628 In dimension zero, any two pairs of points are congruent. (Contributed by Thierry Arnoux, 12-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  ( # `  P )  =  1 )   &    |-  ( ph  ->  D  e.  P )   =>    |-  ( ph  ->  ( A  .-  B )  =  ( C  .-  D ) )
 
Theoremtgbtwndiff 24629* There is always a  c distinct from  B such that  B lies between  A and  c. Theorem 3.14 of [Schwabhauser] p. 32. The condition "the space is of dimension 1 or more" is written here as  2  <_  (
# `  P ) for simplicity. (Contributed by Thierry Arnoux, 23-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  2 
 <_  ( # `  P ) )   =>    |-  ( ph  ->  E. c  e.  P  ( B  e.  ( A I c ) 
 /\  B  =/=  c
 ) )
 
Theoremtgdim01 24630 In geometries of dimension lower than 2, all points are colinear. (Contributed by Thierry Arnoux, 27-Aug-2019.)
 |-  P  =  ( Base `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  -.  GDimTarskiG 2 )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   =>    |-  ( ph  ->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) )
 
Theoremnehash2 24631 The cardinality of a set with two distinct elements. (Contributed by Thierry Arnoux, 27-Aug-2019.)
 |-  ( ph  ->  P  e.  V )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  2  <_  ( # `  P ) )
 
15.2.4  Betweenness and Congruence
 
Theoremtgifscgr 24632 Inner five segment congruence. Take two triangles,  A D C and  E H K, with 
B between  A and  C and  F between  E and  K. If the other components of the triangles are congruent, then so are  B D and  F H. Theorem 4.2 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 24-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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  ->  F  e.  P )   &    |-  ( ph  ->  K  e.  P )   &    |-  ( ph  ->  H  e.  P )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  F  e.  ( E I K ) )   &    |-  ( ph  ->  ( A  .-  C )  =  ( E  .-  K )
 )   &    |-  ( ph  ->  ( B  .-  C )  =  ( F  .-  K ) )   &    |-  ( ph  ->  ( A  .-  D )  =  ( E  .-  H ) )   &    |-  ( ph  ->  ( C  .-  D )  =  ( K  .-  H ) )   =>    |-  ( ph  ->  ( B  .-  D )  =  ( F  .-  H ) )
 
Theoremtgcgrsub 24633 Removing identical parts from the end of a line segment preserves congruence. Theorem 4.3 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  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  ->  F  e.  P )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  E  e.  ( D I F ) )   &    |-  ( ph  ->  ( A  .-  C )  =  ( D  .-  F ) )   &    |-  ( ph  ->  ( B  .-  C )  =  ( E  .-  F ) )   =>    |-  ( ph  ->  ( A  .-  B )  =  ( D  .-  E ) )
 
15.2.5  Congruence of a series of points
 
Syntaxccgrg 24634 Declare the constant for the congruence between shapes relation.
 class cgrG
 
Definitiondf-cgrg 24635* Define the relation congruence bewteen shapes. Definition 4.4 of [Schwabhauser] p. 35. Ideally, we would define this for functions of any set, but we will used words (functions over  NN) in most cases. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |- cgrG  =  ( g  e.  _V  |->  {
 <. a ,  b >.  |  ( ( a  e.  ( ( Base `  g
 )  ^pm  RR )  /\  b  e.  (
 ( Base `  g )  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e. 
 dom  a A. j  e.  dom  a ( ( a `  i ) ( dist `  g )
 ( a `  j
 ) )  =  ( ( b `  i
 ) ( dist `  g
 ) ( b `  j ) ) ) ) } )
 
Theoremiscgrg 24636* The congruence property for sequences of points. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  .~  =  (cgrG `  G )   =>    |-  ( G  e.  V  ->  ( A  .~  B  <->  ( ( A  e.  ( P  ^pm  RR )  /\  B  e.  ( P  ^pm  RR )
 )  /\  ( dom  A  =  dom  B  /\  A. i  e.  dom  A A. j  e.  dom  A ( ( A `  i )  .-  ( A `
  j ) )  =  ( ( B `
  i )  .-  ( B `  j ) ) ) ) ) )
 
Theoremiscgrgd 24637* The property for two sequences  A and  B of points to be congruent. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  D  C_ 
 RR )   &    |-  ( ph  ->  A : D --> P )   &    |-  ( ph  ->  B : D
 --> P )   =>    |-  ( ph  ->  ( A  .~  B  <->  A. i  e.  dom  A
 A. j  e.  dom  A ( ( A `  i )  .-  ( A `
  j ) )  =  ( ( B `
  i )  .-  ( B `  j ) ) ) )
 
Theoremiscgrglt 24638* The property for two sequences  A and  B of points to be congruent, where the congruence is only required for indices verifying a less-than relation. (Contributed by Thierry Arnoux, 7-Oct-2020.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  A : D --> P )   &    |-  ( ph  ->  B : D --> P )   =>    |-  ( ph  ->  ( A  .~  B  <->  A. i  e.  dom  A
 A. j  e.  dom  A ( i  <  j  ->  ( ( A `  i )  .-  ( A `
  j ) )  =  ( ( B `
  i )  .-  ( B `  j ) ) ) ) )
 
Theoremtrgcgrg 24639 The property for two triangles to be congruent to each other. (Contributed by Thierry Arnoux, 3-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  .~  =  (cgrG `  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 E F ">  <->  ( ( A 
 .-  B )  =  ( D  .-  E )  /\  ( B  .-  C )  =  ( E  .-  F )  /\  ( C  .-  A )  =  ( F  .-  D ) ) ) )
 
Theoremtrgcgr 24640 Triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  .~  =  (cgrG `  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 )  =  ( D  .-  E ) )   &    |-  ( ph  ->  ( B  .-  C )  =  ( E  .-  F )
 )   &    |-  ( ph  ->  ( C  .-  A )  =  ( F  .-  D ) )   =>    |-  ( ph  ->  <" A B C ">  .~  <" D E F "> )
 
Theoremercgrg 24641 The shape congruence relation is an equivalence relation. Statement 4.4 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 9-Apr-2019.)
 |-  P  =  ( Base `  G )   =>    |-  ( G  e. TarskiG  ->  (cgrG `  G )  Er  ( P  ^pm  RR ) )
 
Theoremtgcgrxfr 24642* A line segment can be divided at the same place as a congruent line segment is divided. Theorem 4.5 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 9-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  F  e.  P )   &    |-  ( ph  ->  B  e.  ( A I C ) )   &    |-  ( ph  ->  ( A  .-  C )  =  ( D  .-  F ) )   =>    |-  ( ph  ->  E. e  e.  P  ( e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )
 
Theoremcgr3id 24643 Reflexivity law for three-place congruence. (Contributed by Thierry Arnoux, 28-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   =>    |-  ( ph  ->  <" A B C ">  .~  <" A B C "> )
 
Theoremcgr3simp1 24644 Deduce segment congruence from a triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  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 E F "> )   =>    |-  ( ph  ->  ( A  .-  B )  =  ( D  .-  E ) )
 
Theoremcgr3simp2 24645 Deduce segment congruence from a triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  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 E F "> )   =>    |-  ( ph  ->  ( B  .-  C )  =  ( E  .-  F ) )
 
Theoremcgr3simp3 24646 Deduce segment congruence from a triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  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 E F "> )   =>    |-  ( ph  ->  ( C  .-  A )  =  ( F  .-  D ) )
 
Theoremcgr3swap12 24647 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  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 E F "> )   =>    |-  ( ph  ->  <" B A C ">  .~  <" E D F "> )
 
Theoremcgr3swap23 24648 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  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 E F "> )   =>    |-  ( ph  ->  <" A C B ">  .~  <" D F E "> )
 
Theoremcgr3swap13 24649 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 3-Oct-2020.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  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 E F "> )   =>    |-  ( ph  ->  <" C B A ">  .~  <" F E D "> )
 
Theoremcgr3rotr 24650 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 1-Aug-2020.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  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 E F "> )   =>    |-  ( ph  ->  <" C A B ">  .~  <" F D E "> )
 
Theoremcgr3rotl 24651 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 1-Aug-2020.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  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 E F "> )   =>    |-  ( ph  ->  <" B C A ">  .~  <" E F D "> )
 
Theoremtrgcgrcom 24652 Commutative law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  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 E F "> )   =>    |-  ( ph  ->  <" D E F ">  .~  <" A B C "> )
 
Theoremcgr3tr 24653 Transitivity law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  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 E F "> )   &    |-  ( ph  ->  J  e.  P )   &    |-  ( ph  ->  K  e.  P )   &    |-  ( ph  ->  L  e.  P )   &    |-  ( ph  ->  <" D E F ">  .~  <" J K L "> )   =>    |-  ( ph  ->  <" A B C ">  .~  <" J K L "> )
 
Theoremtgbtwnxfr 24654 A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Thierry Arnoux, 27-Apr-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  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 E F "> )   &    |-  ( ph  ->  B  e.  ( A I C ) )   =>    |-  ( ph  ->  E  e.  ( D I F ) )
 
Theoremtgcgr4 24655 Two quadrilaterals to be congruent to each other if one triangle formed by their vertices is, and the additional points are equidistant too. (Contributed by Thierry Arnoux, 8-Oct-2020.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  W  e.  P )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   =>    |-  ( ph  ->  ( <" A B C D ">  .~  <" W X Y Z ">  <->  ( <" A B C ">  .~  <" W X Y ">  /\  (
 ( A  .-  D )  =  ( W  .-  Z )  /\  ( B  .-  D )  =  ( X  .-  Z )  /\  ( C  .-  D )  =  ( Y  .-  Z ) ) ) ) )
 
15.2.6  Motions
 
Syntaxcismt 24656 Declare the constant for the isometry builder.
 class Ismt
 
Definitiondf-ismt 24657* Define the set of isometries between two structures. Definition 4.8 of [Schwabhauser] p. 36. See isismt 24658. (Contributed by Thierry Arnoux, 13-Dec-2019.)
 |- Ismt  =  ( g  e.  _V ,  h  e.  _V  |->  { f  |  ( f : ( Base `  g
 )
 -1-1-onto-> ( Base `  h )  /\  A. a  e.  ( Base `  g ) A. b  e.  ( Base `  g ) ( ( f `  a ) ( dist `  h )
 ( f `  b
 ) )  =  ( a ( dist `  g
 ) b ) ) } )
 
Theoremisismt 24658* Property of being an isometry. Compare with isismty 32197. (Contributed by Thierry Arnoux, 13-Dec-2019.)
 |-  B  =  ( Base `  G )   &    |-  P  =  (
 Base `  H )   &    |-  D  =  ( dist `  G )   &    |-  .-  =  ( dist `  H )   =>    |-  (
 ( G  e.  V  /\  H  e.  W ) 
 ->  ( F  e.  ( GIsmt H )  <->  ( F : B
 -1-1-onto-> P  /\  A. a  e.  B  A. b  e.  B  ( ( F `
  a )  .-  ( F `  b ) )  =  ( a D b ) ) ) )
 
Theoremismot 24659* Property of being an isometry mapping to the same space. In geometry, this is also called a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   =>    |-  ( G  e.  V  ->  ( F  e.  ( GIsmt G )  <->  ( F : P
 -1-1-onto-> P  /\  A. a  e.  P  A. b  e.  P  ( ( F `
  a )  .-  ( F `  b ) )  =  ( a 
 .-  b ) ) ) )
 
Theoremmotcgr 24660 Property of a motion: distances are preserved. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  F  e.  ( GIsmt G ) )   =>    |-  ( ph  ->  ( ( F `  A )  .-  ( F `  B ) )  =  ( A  .-  B ) )
 
Theoremidmot 24661 The identity is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  G  e.  V )   =>    |-  ( ph  ->  (  _I  |`  P )  e.  ( GIsmt G ) )
 
Theoremmotf1o 24662 Motions are bijections. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  F  e.  ( GIsmt G ) )   =>    |-  ( ph  ->  F : P -1-1-onto-> P )
 
Theoremmotcl 24663 Closure of motions. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  F  e.  ( GIsmt G ) )   &    |-  ( ph  ->  A  e.  P )   =>    |-  ( ph  ->  ( F `  A )  e.  P )
 
Theoremmotco 24664 The composition of two motions is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  F  e.  ( GIsmt G ) )   &    |-  ( ph  ->  H  e.  ( GIsmt G ) )   =>    |-  ( ph  ->  ( F  o.  H )  e.  ( GIsmt G ) )
 
Theoremcnvmot 24665 The converse of a motion is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  F  e.  ( GIsmt G ) )   =>    |-  ( ph  ->  `' F  e.  ( GIsmt G ) )
 
Theoremmotplusg 24666* The operation for motions is their composition. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  I  =  { <. (
 Base `  ndx ) ,  ( GIsmt G )
 >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( GIsmt G ) ,  g  e.  ( GIsmt G )  |->  ( f  o.  g ) )
 >. }   &    |-  ( ph  ->  F  e.  ( GIsmt G ) )   &    |-  ( ph  ->  H  e.  ( GIsmt G ) )   =>    |-  ( ph  ->  ( F ( +g  `  I
 ) H )  =  ( F  o.  H ) )
 
Theoremmotgrp 24667* The motions of a geometry form a group with respect to function composition, called the Isometry group. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  I  =  { <. (
 Base `  ndx ) ,  ( GIsmt G )
 >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( GIsmt G ) ,  g  e.  ( GIsmt G )  |->  ( f  o.  g ) )
 >. }   =>    |-  ( ph  ->  I  e.  Grp )
 
Theoremmotcgrg 24668* Property of a motion: distances are preserved. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  I  =  { <. (
 Base `  ndx ) ,  ( GIsmt G )
 >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( GIsmt G ) ,  g  e.  ( GIsmt G )  |->  ( f  o.  g ) )
 >. }   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  T  e. Word  P )   &    |-  ( ph  ->  F  e.  ( GIsmt G ) )   =>    |-  ( ph  ->  ( F  o.  T ) 
 .~  T )
 
Theoremmotcgr3 24669 Property of a motion: distances are preserved, special case of triangles. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  .~  =  (cgrG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  =  ( H `  A ) )   &    |-  ( ph  ->  E  =  ( H `  B ) )   &    |-  ( ph  ->  F  =  ( H `  C ) )   &    |-  ( ph  ->  H  e.  ( GIsmt G ) )   =>    |-  ( ph  ->  <" A B C ">  .~  <" D E F "> )
 
15.2.7  Colinearity
 
Theoremtglng 24670* Lines of a Tarski Geometry. This relates to both Definition 4.10 of [Schwabhauser] p. 36. and Definition 6.14 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 28-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   =>    |-  ( G  e. TarskiG  ->  L  =  ( x  e.  P ,  y  e.  ( P  \  { x }
 )  |->  { z  e.  P  |  ( z  e.  ( x I y )  \/  x  e.  ( z I y )  \/  y  e.  ( x I z ) ) } ) )
 
Theoremtglnfn 24671 Lines as functions. (Contributed by Thierry Arnoux, 25-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   =>    |-  ( G  e. TarskiG  ->  L  Fn  ( ( P  X.  P )  \  _I  )
 )
 
Theoremtglnunirn 24672 Lines are sets of points. (Contributed by Thierry Arnoux, 25-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   =>    |-  ( G  e. TarskiG  ->  U. ran  L 
 C_  P )
 
Theoremtglnpt 24673 Lines are sets of points. (Contributed by Thierry Arnoux, 17-Oct-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  ran  L )   &    |-  ( ph  ->  X  e.  A )   =>    |-  ( ph  ->  X  e.  P )
 
Theoremtglngne 24674 It takes two different points to form a line. (Contributed by Thierry Arnoux, 6-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.  ( X L Y ) )   =>    |-  ( ph  ->  X  =/=  Y )
 
Theoremtglngval 24675* The line going through points  X and  Y. (Contributed by Thierry Arnoux, 28-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  ( X L Y )  =  { z  e.  P  |  ( z  e.  ( X I Y )  \/  X  e.  ( z I Y )  \/  Y  e.  ( X I z ) ) } )
 
Theoremtglnssp 24676 Lines are subset of the geometry base set. That is, lines are sets of points. (Contributed by Thierry Arnoux, 17-May-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  ( X L Y )  C_  P )
 
Theoremtgellng 24677 Property of lying on the line going through points  X and 
Y. Definition 4.10 of [Schwabhauser] p. 36. We choose the notation  Z  e.  ( X (LineG `  G
) Y ) instead of "colinear" because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 28-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  L  =  (LineG `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Z  e.  P )   =>    |-  ( ph  ->  ( Z  e.  ( X L Y )  <->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
 
Theoremtgcolg 24678 We choose the notation  ( Z  e.  ( X L Y )  \/  X  =  Y ) instead of "colinear" in order to avoid defining an additional symbol for colinearity because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 25-May-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 )  <-> 
 ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
 
Theorembtwncolg1 24679 Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-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 I Y ) )   =>    |-  ( ph  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )
 
Theorembtwncolg2 24680 Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-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  ->  X  e.  ( Z I Y ) )   =>    |-  ( ph  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )
 
Theorembtwncolg3 24681 Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-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  ->  Y  e.  ( X I Z ) )   =>    |-  ( ph  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )
 
Theoremcolcom 24682 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 24683 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 24684 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 24685 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 24686 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 24687 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 24688 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 24689 If there are 3 non-colinear points, dimension must be 2 or more. tglowdim2l 24774 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 24690 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 24691* 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 24692 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 24693 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 24694 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 24695 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 24696 Lemma for tgbtwnconn1 24699. (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 24697 Lemma for tgbtwnconn1 24699. (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 24698 Lemma for tgbtwnconn1 24699. (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 24699 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 24700 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 ) ) )
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