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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcgr3permute5 25701 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) 
 /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >.
 >.Cgr3 <. D ,  <. E ,  F >. >.  <->  <. C ,  <. B ,  A >. >.Cgr3 <. F ,  <. E ,  D >. >.
 ) )
 
Theoremcgr3tr4 25702 Transitivity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N ) 
 /\  I  e.  ( EE `  N ) ) ) )  ->  (
 ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. G ,  <. H ,  I >. >.
 )  ->  <. D ,  <. E ,  F >. >.Cgr3 <. G ,  <. H ,  I >. >. ) )
 
Theoremcgr3com 25703 Commutativity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) 
 /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >.
 >.Cgr3 <. D ,  <. E ,  F >. >.  <->  <. D ,  <. E ,  F >. >.Cgr3 <. A ,  <. B ,  C >. >.
 ) )
 
Theoremcgr3rflx 25704 Identity law for three-place congruence. (Contributed by Scott Fenton, 6-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >. )
 
Theoremcgrxfr 25705* A line segment can be divided at the same place as a congruent line segment is divided. Theorem 4.5 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )  ->  (
 ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. )  ->  E. e  e.  ( EE `  N ) ( e  Btwn  <. D ,  F >.  /\ 
 <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )
 
Theorembtwnxfr 25706 A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 4-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) 
 /\  F  e.  ( EE `  N ) ) )  ->  ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.
 )  ->  E  Btwn  <. D ,  F >. ) )
 
Theoremcolinrel 25707 Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Rel  Colinear
 
Theorembrcolinear2 25708* Alternate colinearity binary relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( Q  e.  V  /\  R  e.  W ) 
 ->  ( P  Colinear  <. Q ,  R >. 
 <-> 
 E. n  e.  NN  ( ( P  e.  ( EE `  n ) 
 /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) )  /\  ( P  Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) ) ) )
 
Theorembrcolinear 25709 The binary relationship form of the colinearity predicate. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Colinear  <. B ,  C >. 
 <->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )
 
Theoremcolinearex 25710 The colinear predicate exists. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Colinear  e.  _V
 
Theoremcolineardim1 25711 If  A is colinear with  B and  C, then  A is in the same space as  B. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE
 `  N )  /\  C  e.  W )
 )  ->  ( A  Colinear  <. B ,  C >.  ->  A  e.  ( EE `  N ) ) )
 
Theoremcolinearperm1 25712 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Colinear  <. B ,  C >. 
 <->  A  Colinear  <. C ,  B >. ) )
 
Theoremcolinearperm3 25713 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Colinear  <. B ,  C >. 
 <->  B  Colinear  <. C ,  A >. ) )
 
Theoremcolinearperm2 25714 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Colinear  <. B ,  C >. 
 <->  B  Colinear  <. A ,  C >. ) )
 
Theoremcolinearperm4 25715 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Colinear  <. B ,  C >. 
 <->  C  Colinear  <. A ,  B >. ) )
 
Theoremcolinearperm5 25716 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Colinear  <. B ,  C >. 
 <->  C  Colinear  <. B ,  A >. ) )
 
Theoremcolineartriv1 25717 Trivial case of colinearity. Theorem 4.12 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  A 
 Colinear 
 <. A ,  B >. )
 
Theoremcolineartriv2 25718 Trivial case of colinearity. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  A 
 Colinear 
 <. B ,  B >. )
 
Theorembtwncolinear1 25719 Betweenness implies colinearity. (Contributed by Scott Fenton, 7-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( C  Btwn  <. A ,  B >.  ->  A  Colinear  <. B ,  C >. ) )
 
Theorembtwncolinear2 25720 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( C  Btwn  <. A ,  B >.  ->  A  Colinear  <. C ,  B >. ) )
 
Theorembtwncolinear3 25721 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( C  Btwn  <. A ,  B >.  ->  B  Colinear  <. A ,  C >. ) )
 
Theorembtwncolinear4 25722 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( C  Btwn  <. A ,  B >.  ->  B  Colinear  <. C ,  A >. ) )
 
Theorembtwncolinear5 25723 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( C  Btwn  <. A ,  B >.  ->  C  Colinear  <. A ,  B >. ) )
 
Theorembtwncolinear6 25724 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( C  Btwn  <. A ,  B >.  ->  C  Colinear  <. B ,  A >. ) )
 
Theoremcolinearxfr 25725 Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) 
 /\  F  e.  ( EE `  N ) ) )  ->  ( ( B 
 Colinear 
 <. A ,  C >.  /\ 
 <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  E 
 Colinear 
 <. D ,  F >. ) )
 
Theoremlineext 25726* Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  (
 ( A  Colinear  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. )  ->  E. f  e.  ( EE `  N ) <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >.
 ) )
 
Theorembrofs2 25727 Change some conditions for outer five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013.)
 |-  (
 ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  G  e.  ( EE `  N ) 
 /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  B >. , 
 <. C ,  D >. >.  OuterFiveSeg  <. <. E ,  F >. , 
 <. G ,  H >. >.  <->  ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) ) ) )
 
Theorembrifs2 25728 Change some conditions for inner five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013.)
 |-  (
 ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  G  e.  ( EE `  N ) 
 /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  B >. , 
 <. C ,  D >. >.  InnerFiveSeg  <. <. E ,  F >. , 
 <. G ,  H >. >.  <->  ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. C ,  D >.Cgr <. G ,  H >. ) ) ) )
 
Theorembrfs 25729 Binary relationship form of the general five segment predicate. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  G  e.  ( EE `  N ) 
 /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  B >. , 
 <. C ,  D >. >.  FiveSeg  <. <. E ,  F >. , 
 <. G ,  H >. >.  <->  ( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr
 <. E ,  H >.  /\ 
 <. B ,  D >.Cgr <. F ,  H >. ) ) ) )
 
Theoremfscgr 25730 Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  (
 ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  G  e.  ( EE `  N ) 
 /\  H  e.  ( EE `  N ) ) )  ->  ( ( <.
 <. A ,  B >. , 
 <. C ,  D >. >.  FiveSeg  <. <. E ,  F >. , 
 <. G ,  H >. >.  /\  A  =/=  B ) 
 ->  <. C ,  D >.Cgr
 <. G ,  H >. ) )
 
Theoremlinecgr 25731 Congruence rule for lines. Theorem 4.17 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N ) ) )  ->  (
 ( ( A  =/=  B 
 /\  A  Colinear  <. B ,  C >. )  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\ 
 <. B ,  P >.Cgr <. B ,  Q >. ) )  ->  <. C ,  P >.Cgr <. C ,  Q >. ) )
 
Theoremlinecgrand 25732 Deduction form of linecgr 25731. (Contributed by Scott Fenton, 14-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  P  e.  ( EE `  N ) )   &    |-  ( ph  ->  Q  e.  ( EE `  N ) )   &    |-  (
 ( ph  /\  ps )  ->  A  =/=  B )   &    |-  ( ( ph  /\  ps )  ->  A  Colinear  <. B ,  C >. )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  P >.Cgr <. A ,  Q >. )   &    |-  ( ( ph  /\ 
 ps )  ->  <. B ,  P >.Cgr <. B ,  Q >. )   =>    |-  ( ( ph  /\  ps )  ->  <. C ,  P >.Cgr
 <. C ,  Q >. )
 
Theoremlineid 25733 Identity law for points on lines. Theorem 4.18 of [Schwabhauser] p. 38. (Contributed by Scott Fenton, 7-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( ( A  =/=  B  /\  A  Colinear  <. B ,  C >. ) 
 /\  ( <. A ,  C >.Cgr <. A ,  D >.  /\  <. B ,  C >.Cgr
 <. B ,  D >. ) )  ->  C  =  D ) )
 
Theoremidinside 25734 Law for finding a point inside a segment. Theorem 4.19 of [Schwabhauser] p. 38. (Contributed by Scott Fenton, 7-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( C  Btwn  <. A ,  B >.  /\ 
 <. A ,  C >.Cgr <. A ,  D >.  /\ 
 <. B ,  C >.Cgr <. B ,  D >. ) 
 ->  C  =  D ) )
 
Theoremendofsegid 25735 If  A,  B, and  C fall in order on a line, and  A B and  A C are congruent, then  C  =  B. (Contributed by Scott Fenton, 7-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( ( B  Btwn  <. A ,  C >.  /\ 
 <. A ,  C >.Cgr <. A ,  B >. ) 
 ->  C  =  B ) )
 
Theoremendofsegidand 25736 Deduction form of endofsegid 25735. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  C  Btwn  <. A ,  B >. )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. A ,  C >. )   =>    |-  ( ( ph  /\  ps )  ->  B  =  C )
 
19.7.45  Connectivity of betweenness
 
Theorembtwnconn1lem1 25737 Lemma for btwnconn1 25751. The next several lemmas introduce various properties of hypothetical points that end up eliminating alternatives to connectivity. We begin by showing a congruence property of those hypothetical points. (Contributed by Scott Fenton, 8-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  X  e.  ( EE `  N ) ) )  /\  ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
 <. D ,  c >.Cgr <. C ,  D >. ) 
 /\  ( C  Btwn  <. A ,  d >.  /\ 
 <. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  X >.  /\  <. d ,  X >.Cgr
 <. D ,  B >. ) ) ) )  ->  <. B ,  c >.Cgr <. X ,  C >. )
 
Theorembtwnconn1lem2 25738 Lemma for btwnconn1 25751. Now, we show that two of the hypotheticals we introduced in the first lemma are identical. (Contributed by Scott Fenton, 8-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  X  e.  ( EE `  N ) ) )  /\  ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
 <. D ,  c >.Cgr <. C ,  D >. ) 
 /\  ( C  Btwn  <. A ,  d >.  /\ 
 <. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  X >.  /\  <. d ,  X >.Cgr
 <. D ,  B >. ) ) ) )  ->  X  =  b )
 
Theorembtwnconn1lem3 25739 Lemma for btwnconn1 25751. Establish the next congruence in the series. (Contributed by Scott Fenton, 8-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N ) ) ) 
 /\  ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\  <. D ,  c >.Cgr <. C ,  D >. )  /\  ( C 
 Btwn  <. A ,  d >.  /\  <. C ,  d >.Cgr
 <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  b >.  /\  <. d ,  b >.Cgr
 <. D ,  B >. ) ) ) )  ->  <. B ,  d >.Cgr <.
 b ,  D >. )
 
Theorembtwnconn1lem4 25740 Lemma for btwnconn1 25751. Assuming  C  =/=  c, we now attempt to force  D  =  d from here out via a series of congruences. (Contributed by Scott Fenton, 8-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N ) ) ) 
 /\  ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
 <. D ,  c >.Cgr <. C ,  D >. ) 
 /\  ( C  Btwn  <. A ,  d >.  /\ 
 <. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  b >.  /\  <. d ,  b >.Cgr
 <. D ,  B >. ) ) ) )  ->  <. d ,  c >.Cgr <. D ,  C >. )
 
Theorembtwnconn1lem5 25741 Lemma for btwnconn1 25751. Now, we introduce  E, the intersection of  C c and  D d. We begin by showing that it is the midpoint of  C and  c (Contributed by Scott Fenton, 8-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  /\  ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
 <. D ,  c >.Cgr <. C ,  D >. ) 
 /\  ( C  Btwn  <. A ,  d >.  /\ 
 <. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  b >.  /\  <. d ,  b >.Cgr
 <. D ,  B >. ) ) )  /\  ( E  Btwn  <. C ,  c >.  /\  E  Btwn  <. D ,  d >. ) ) ) 
 ->  <. E ,  C >.Cgr
 <. E ,  c >. )
 
Theorembtwnconn1lem6 25742 Lemma for btwnconn1 25751. Next, we show that  E is the midpoint of  D and  d (Contributed by Scott Fenton, 8-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  /\  ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
 <. D ,  c >.Cgr <. C ,  D >. ) 
 /\  ( C  Btwn  <. A ,  d >.  /\ 
 <. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  b >.  /\  <. d ,  b >.Cgr
 <. D ,  B >. ) ) )  /\  ( E  Btwn  <. C ,  c >.  /\  E  Btwn  <. D ,  d >. ) ) ) 
 ->  <. E ,  D >.Cgr
 <. E ,  d >. )
 
Theorembtwnconn1lem7 25743 Lemma for btwnconn1 25751. Under our assumptions,  C and  d are distinct. (Contributed by Scott Fenton, 8-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  /\  ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
 <. D ,  c >.Cgr <. C ,  D >. ) 
 /\  ( C  Btwn  <. A ,  d >.  /\ 
 <. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  b >.  /\  <. d ,  b >.Cgr
 <. D ,  B >. ) ) )  /\  ( E  Btwn  <. C ,  c >.  /\  E  Btwn  <. D ,  d >. ) ) ) 
 ->  C  =/=  d )
 
Theorembtwnconn1lem8 25744 Lemma for btwnconn1 25751. Now, we introduce the last three points used in the construction:  P,  Q, and  R will turn out to be equal further down, and will provide us with the key to the final statement. We begin by establishing congruence of  R P and  E d (Contributed by Scott Fenton, 8-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) 
 /\  c  e.  ( EE `  N ) ) 
 /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) ) 
 /\  ( P  e.  ( EE `  N ) 
 /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) ) ) 
 /\  ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\  <. D ,  c >.Cgr <. C ,  D >. )  /\  ( C 
 Btwn  <. A ,  d >.  /\  <. C ,  d >.Cgr
 <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  b >.  /\  <. d ,  b >.Cgr
 <. D ,  B >. ) ) )  /\  (
 ( E  Btwn  <. C ,  c >.  /\  E  Btwn  <. D ,  d >. ) 
 /\  ( ( C 
 Btwn  <. c ,  P >.  /\  <. C ,  P >.Cgr
 <. C ,  d >. ) 
 /\  ( C  Btwn  <.
 d ,  R >.  /\ 
 <. C ,  R >.Cgr <. C ,  E >. ) 
 /\  ( R  Btwn  <. P ,  Q >.  /\ 
 <. R ,  Q >.Cgr <. R ,  P >. ) ) ) ) ) 
 ->  <. R ,  P >.Cgr
 <. E ,  d >. )
 
Theorembtwnconn1lem9 25745 Lemma for btwnconn1 25751. Now, a quick use of transitivity to establish congruence on  R Q and  E D (Contributed by Scott Fenton, 8-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) 
 /\  c  e.  ( EE `  N ) ) 
 /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) ) 
 /\  ( P  e.  ( EE `  N ) 
 /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) ) ) 
 /\  ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\  <. D ,  c >.Cgr <. C ,  D >. )  /\  ( C 
 Btwn  <. A ,  d >.  /\  <. C ,  d >.Cgr
 <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  b >.  /\  <. d ,  b >.Cgr
 <. D ,  B >. ) ) )  /\  (
 ( E  Btwn  <. C ,  c >.  /\  E  Btwn  <. D ,  d >. ) 
 /\  ( ( C 
 Btwn  <. c ,  P >.  /\  <. C ,  P >.Cgr
 <. C ,  d >. ) 
 /\  ( C  Btwn  <.
 d ,  R >.  /\ 
 <. C ,  R >.Cgr <. C ,  E >. ) 
 /\  ( R  Btwn  <. P ,  Q >.  /\ 
 <. R ,  Q >.Cgr <. R ,  P >. ) ) ) ) ) 
 ->  <. R ,  Q >.Cgr
 <. E ,  D >. )
 
Theorembtwnconn1lem10 25746 Lemma for btwnconn1 25751. Now we establish a congruence that will give us  D  =  d when we compute  P  =  Q later on. (Contributed by Scott Fenton, 8-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) 
 /\  c  e.  ( EE `  N ) ) 
 /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) ) 
 /\  ( P  e.  ( EE `  N ) 
 /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) ) ) 
 /\  ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\  <. D ,  c >.Cgr <. C ,  D >. )  /\  ( C 
 Btwn  <. A ,  d >.  /\  <. C ,  d >.Cgr
 <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  b >.  /\  <. d ,  b >.Cgr
 <. D ,  B >. ) ) )  /\  (
 ( E  Btwn  <. C ,  c >.  /\  E  Btwn  <. D ,  d >. ) 
 /\  ( ( C 
 Btwn  <. c ,  P >.  /\  <. C ,  P >.Cgr
 <. C ,  d >. ) 
 /\  ( C  Btwn  <.
 d ,  R >.  /\ 
 <. C ,  R >.Cgr <. C ,  E >. ) 
 /\  ( R  Btwn  <. P ,  Q >.  /\ 
 <. R ,  Q >.Cgr <. R ,  P >. ) ) ) ) ) 
 ->  <. d ,  D >.Cgr
 <. P ,  Q >. )
 
Theorembtwnconn1lem11 25747 Lemma for btwnconn1 25751. Now, we establish that  D and  Q are equidistant from  C (Contributed by Scott Fenton, 8-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) 
 /\  c  e.  ( EE `  N ) ) 
 /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) ) 
 /\  ( P  e.  ( EE `  N ) 
 /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) ) ) 
 /\  ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\  <. D ,  c >.Cgr <. C ,  D >. )  /\  ( C 
 Btwn  <. A ,  d >.  /\  <. C ,  d >.Cgr
 <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  b >.  /\  <. d ,  b >.Cgr
 <. D ,  B >. ) ) )  /\  (
 ( E  Btwn  <. C ,  c >.  /\  E  Btwn  <. D ,  d >. ) 
 /\  ( ( C 
 Btwn  <. c ,  P >.  /\  <. C ,  P >.Cgr
 <. C ,  d >. ) 
 /\  ( C  Btwn  <.
 d ,  R >.  /\ 
 <. C ,  R >.Cgr <. C ,  E >. ) 
 /\  ( R  Btwn  <. P ,  Q >.  /\ 
 <. R ,  Q >.Cgr <. R ,  P >. ) ) ) ) ) 
 ->  <. D ,  C >.Cgr
 <. Q ,  C >. )
 
Theorembtwnconn1lem12 25748 Lemma for btwnconn1 25751. Using a long string of invocations of linecgr 25731, we show that  D  =  d. (Contributed by Scott Fenton, 9-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) 
 /\  c  e.  ( EE `  N ) ) 
 /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) ) 
 /\  ( P  e.  ( EE `  N ) 
 /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) ) ) 
 /\  ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\  <. D ,  c >.Cgr <. C ,  D >. )  /\  ( C 
 Btwn  <. A ,  d >.  /\  <. C ,  d >.Cgr
 <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  b >.  /\  <. d ,  b >.Cgr
 <. D ,  B >. ) ) )  /\  (
 ( E  Btwn  <. C ,  c >.  /\  E  Btwn  <. D ,  d >. ) 
 /\  ( ( C 
 Btwn  <. c ,  P >.  /\  <. C ,  P >.Cgr
 <. C ,  d >. ) 
 /\  ( C  Btwn  <.
 d ,  R >.  /\ 
 <. C ,  R >.Cgr <. C ,  E >. ) 
 /\  ( R  Btwn  <. P ,  Q >.  /\ 
 <. R ,  Q >.Cgr <. R ,  P >. ) ) ) ) ) 
 ->  D  =  d )
 
Theorembtwnconn1lem13 25749 Lemma for btwnconn1 25751. Begin back-filling and eliminating hypotheses. (Contributed by Scott Fenton, 9-Oct-2013.)
 |-  (
 ( ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 /\  ( ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) 
 /\  c  e.  ( EE `  N ) ) 
 /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  (
 ( ( A  =/=  B 
 /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
 <. D ,  c >.Cgr <. C ,  D >. ) 
 /\  ( C  Btwn  <. A ,  d >.  /\ 
 <. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
 Btwn  <. A ,  b >.  /\  <. d ,  b >.Cgr
 <. D ,  B >. ) ) ) )  ->  ( C  =  c  \/  D  =  d ) )
 
Theorembtwnconn1lem14 25750 Lemma for btwnconn1 25751. Final statement of the theorem when  B  =/=  C. (Contributed by Scott Fenton, 9-Oct-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) )  /\  ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) )
 
Theorembtwnconn1 25751 Connectitivy law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. (Contributed by Scott Fenton, 9-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( A  =/=  B 
 /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) 
 ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) ) )
 
Theorembtwnconn2 25752 Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( A  =/=  B 
 /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) 
 ->  ( C  Btwn  <. B ,  D >.  \/  D  Btwn  <. B ,  C >. ) ) )
 
Theorembtwnconn3 25753 Inner connectivity law for betweenness. Theorem 5.3 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. )  ->  ( B  Btwn  <. A ,  C >.  \/  C  Btwn  <. A ,  B >. ) ) )
 
Theoremmidofsegid 25754 If two points fall in the same place in the middle of a segment, then they are identical. (Contributed by Scott Fenton, 16-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) ) 
 ->  ( ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\  <. A ,  D >.Cgr <. A ,  E >. )  ->  D  =  E ) )
 
Theoremsegcon2 25755* Generalization of axsegcon 25582. This time, we generate an endpoint for a segment on the ray  Q A congruent to  B C and starting at  Q, as opposed to axsegcon 25582, where the segment starts at  A (Contributed by Scott Fenton, 14-Oct-2013.) (Removed unneeded inequality, 15-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) ) 
 ->  E. x  e.  ( EE `  N ) ( ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) 
 /\  <. Q ,  x >.Cgr
 <. B ,  C >. ) )
 
19.7.46  Segment less than or equal to
 
Syntaxcsegle 25756 Declare the constant for the segment less than or equal to relationship.
 class  Seg<_
 
Definitiondf-segle 25757* Define the segment length comparison relationship. This relationship expresses that the segment 
A B is no longer than  C D. In this section, we establish various properties of this relationship showing that it is a transitive, reflexive relationship on pairs of points that is substitutive under congruence. Definition 5.4 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 11-Oct-2013.)
 |-  Seg<_  =  { <. p ,  q >.  | 
 E. n  e.  NN  E. a  e.  ( EE
 `  n ) E. b  e.  ( EE `  n ) E. c  e.  ( EE `  n ) E. d  e.  ( EE `  n ) ( p  =  <. a ,  b >.  /\  q  = 
 <. c ,  d >.  /\ 
 E. y  e.  ( EE `  n ) ( y  Btwn  <. c ,  d >.  /\  <. a ,  b >.Cgr <. c ,  y >. ) ) }
 
Theorembrsegle 25758* Binary relationship form of the segment comparison relationship. (Contributed by Scott Fenton, 11-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >. 
 Seg<_ 
 <. C ,  D >.  <->  E. y  e.  ( EE `  N ) ( y 
 Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr
 <. C ,  y >. ) ) )
 
Theorembrsegle2 25759* Alternate characterization of segment comparison. Theorem 5.5 of [Schwabhauser] p. 41-42. (Contributed by Scott Fenton, 11-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >. 
 Seg<_ 
 <. C ,  D >.  <->  E. x  e.  ( EE `  N ) ( B 
 Btwn  <. A ,  x >.  /\  <. A ,  x >.Cgr
 <. C ,  D >. ) ) )
 
Theoremseglecgr12im 25760 Substitution law for segment comparison under congruence. Theorem 5.6 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
 |-  (
 ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  G  e.  ( EE `  N ) 
 /\  H  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
 <. C ,  D >.Cgr <. G ,  H >.  /\ 
 <. A ,  B >.  Seg<_  <. C ,  D >. ) 
 ->  <. E ,  F >. 
 Seg<_ 
 <. G ,  H >. ) )
 
Theoremseglecgr12 25761 Substitution law for segment comparison under congruence. Biconditional version. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  G  e.  ( EE `  N ) 
 /\  H  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
 <. C ,  D >.Cgr <. G ,  H >. ) 
 ->  ( <. A ,  B >. 
 Seg<_ 
 <. C ,  D >.  <->  <. E ,  F >.  Seg<_  <. G ,  H >. ) ) )
 
Theoremseglerflx 25762 Segment comparison is reflexive. Theorem 5.7 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.  Seg<_  <. A ,  B >. )
 
Theoremseglemin 25763 Any segment is at least as long as a degenerate segment. Theorem 5.11 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  <. A ,  A >.  Seg<_  <. B ,  C >. )
 
Theoremsegletr 25764 Segment less than is transitive. Theorem 5.8 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) 
 /\  F  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.  Seg<_  <. C ,  D >.  /\ 
 <. C ,  D >.  Seg<_  <. E ,  F >. ) 
 ->  <. A ,  B >. 
 Seg<_ 
 <. E ,  F >. ) )
 
Theoremsegleantisym 25765 Antisymmetry law for segment comparison. Theorem 5.9 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( <. A ,  B >.  Seg<_  <. C ,  D >.  /\  <. C ,  D >. 
 Seg<_ 
 <. A ,  B >. ) 
 ->  <. A ,  B >.Cgr
 <. C ,  D >. ) )
 
Theoremseglelin 25766 Linearity law for segment comparison. Theorem 5.10 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >. 
 Seg<_ 
 <. C ,  D >.  \/ 
 <. C ,  D >.  Seg<_  <. A ,  B >. ) )
 
Theorembtwnsegle 25767 If  B falls between  A and  C, then  A B is no longer than  A C. (Contributed by Scott Fenton, 16-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( B  Btwn  <. A ,  C >.  ->  <. A ,  B >. 
 Seg<_ 
 <. A ,  C >. ) )
 
Theoremcolinbtwnle 25768 Given three colinear points  A,  B, and  C,  B falls in the middle iff the two segments to 
B are no longer than  A C. Theorem 5.12 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Colinear  <. B ,  C >.  ->  ( B  Btwn  <. A ,  C >.  <->  ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\ 
 <. B ,  C >.  Seg<_  <. A ,  C >. ) ) ) )
 
19.7.47  Outside of relationship
 
Syntaxcoutsideof 25769 Declare the syntax for the outside of constant.
 class OutsideOf
 
Definitiondf-outsideof 25770 The outside of relationship. This relationship expresses that  P,  A, and  B fall on a line, but  P is not on the segment  A B. This definition is taken from theorem 6.4 of [Schwabhauser] p. 43, since it requires no dummy variables. (Contributed by Scott Fenton, 17-Oct-2013.)
 |- OutsideOf  =  (  Colinear  \  Btwn  )
 
Theorembroutsideof 25771 Binary relationship form of OutsideOf. Theorem 6.4 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( POutsideOf
 <. A ,  B >.  <->  ( P 
 Colinear 
 <. A ,  B >.  /\ 
 -.  P  Btwn  <. A ,  B >. ) )
 
Theorembroutsideof2 25772 Alternate form of OutsideOf. Definition 6.1 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >.  <-> 
 ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
 
Theoremoutsidene1 25773 Outsideness implies inequality. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >.  ->  A  =/=  P ) )
 
Theoremoutsidene2 25774 Outsideness implies inequality. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >.  ->  B  =/=  P ) )
 
Theorembtwnoutside 25775 A principle linking outsideness to betweenness. Theorem 6.2 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) ) 
 ->  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P )  /\  P  Btwn  <. A ,  C >. ) 
 ->  ( P  Btwn  <. B ,  C >. 
 <->  POutsideOf <. A ,  B >. ) ) )
 
Theorembroutsideof3 25776* Characterization of outsideness in terms of relationship to a fourth point. Theorem 6.3 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >.  <-> 
 ( A  =/=  P  /\  B  =/=  P  /\  E. c  e.  ( EE
 `  N ) ( c  =/=  P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B ,  c >. ) ) ) )
 
Theoremoutsideofrflx 25777 Reflexitivity of outsideness. Theorem 6.5 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  P  e.  ( EE
 `  N )  /\  A  e.  ( EE `  N ) )  ->  ( A  =/=  P  ->  POutsideOf <. A ,  A >. ) )
 
Theoremoutsideofcom 25778 Commutitivity law for outsideness. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >.  <->  POutsideOf
 <. B ,  A >. ) )
 
Theoremoutsideoftr 25779 Transitivity law for outsideness. Theorem 6.7 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) ) 
 ->  ( ( POutsideOf <. A ,  B >.  /\  POutsideOf <. B ,  C >. )  ->  POutsideOf <. A ,  C >. ) )
 
Theoremoutsideofeq 25780 Uniqueness law for OutsideOf. Analog of segconeq 25660. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N ) 
 /\  Y  e.  ( EE `  N ) ) )  ->  ( (
 ( AOutsideOf <. X ,  R >.  /\  <. A ,  X >.Cgr
 <. B ,  C >. ) 
 /\  ( AOutsideOf <. Y ,  R >.  /\  <. A ,  Y >.Cgr <. B ,  C >. ) )  ->  X  =  Y ) )
 
Theoremoutsideofeu 25781* Given a non-degenerate ray, there is a unique point congruent to the segment  B C lying on the ray  A R. Theorem 6.11 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 23-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) ) 
 ->  ( ( R  =/=  A 
 /\  B  =/=  C )  ->  E! x  e.  ( EE `  N ) ( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr <. B ,  C >. ) ) )
 
Theoremoutsidele 25782 Relate OutsideOf to  Seg<_. Theorem 6.13 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >.  ->  ( <. P ,  A >.  Seg<_  <. P ,  B >.  <->  A  Btwn  <. P ,  B >. ) ) )
 
Theoremoutsideofcol 25783 Outside of implies colinearity. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( POutsideOf
 <. Q ,  R >.  ->  P 
 Colinear 
 <. Q ,  R >. )
 
19.7.48  Lines and Rays
 
Syntaxcline2 25784 Declare the constant for the line function.
 class Line
 
Syntaxcray 25785 Declare the constant for the ray function.
 class Ray
 
Syntaxclines2 25786 Declare the constant for the set of all lines.
 class LinesEE
 
Definitiondf-line2 25787* Define the Line function. This function generates the line passing through the distinct points  a and  b. Adapted from definition 6.14 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 25-Oct-2013.)
 |- Line  =  { <.
 <. a ,  b >. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
 )  /\  l  =  [ <. a ,  b >. ] `'  Colinear  ) }
 
Definitiondf-ray 25788* Define the Ray function. This function generates the set of all points that lie on the ray starting at  p and passing through  a. Definition 6.8 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 21-Oct-2013.)
 |- Ray  =  { <.
 <. p ,  a >. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
 )  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) }
 
Definitiondf-lines2 25789 Define the set of all lines. Definition 6.14, part 2 of [Schwabhauser] p. 45. See ellines 25802 for membership. (Contributed by Scott Fenton, 28-Oct-2013.)
 |- LinesEE  =  ran Line
 
Theoremfunray 25790 Show that the Ray relationship is a function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun Ray
 
Theoremfvray 25791* Calculate the value of the Ray function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  ( PRay A )  =  { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. } )
 
Theoremfunline 25792 Show that the Line relationship is a function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun Line
 
Theoremlinedegen 25793 When Line is applied with the same argument, the result is the empty set. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ALine A )  =  (/)
 
Theoremfvline 25794* Calculate the value of the Line function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  ( ALine B )  =  { x  |  x  Colinear  <. A ,  B >. } )
 
Theoremliness 25795 A line is a subset of the space its two points lie in. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  ( ALine B )  C_  ( EE `  N ) )
 
Theoremfvline2 25796* Alternate definition of a line. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  ( ALine B )  =  { x  e.  ( EE `  N )  |  x  Colinear  <. A ,  B >. } )
 
Theoremlineunray 25797 A line is composed of a point and the two rays emerging from it. Theorem 6.15 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  ( P  Btwn  <. Q ,  R >.  ->  ( PLine Q )  =  ( (
 ( PRay Q )  u.  { P }
 )  u.  ( PRay R ) ) ) )
 
Theoremlineelsb2 25798 If  S lies on  P Q, then 
P Q  =  P S. Theorem 6.16 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  ( S  e.  ( EE `  N )  /\  P  =/=  S ) )  ->  ( S  e.  ( PLine Q )  ->  ( PLine Q )  =  ( PLine S ) ) )
 
Theoremlinerflx1 25799 Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  P  e.  ( PLine Q ) )
 
Theoremlinecom 25800 Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  ( PLine Q )  =  ( QLine P ) )
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