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Theorem List for Metamath Proof Explorer - 24101-24200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembtwncolinear2 24101 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 24102 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 24103 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 24104 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 24105 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 24106 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 24107* 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 24108 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 24109 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 24110 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 24111 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 24112 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 24113 Deduction form of linecgr 24112. (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 24114 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 24115 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 24116 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 24117 Deduction form of endofsegid 24116. (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 )
 
18.7.36  Connectivity of betweenness
 
Theorembtwnconn1lem1 24118 Lemma for btwnconn1 24132. 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 24119 Lemma for btwnconn1 24132. 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 24120 Lemma for btwnconn1 24132. 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 24121 Lemma for btwnconn1 24132. 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 24122 Lemma for btwnconn1 24132. 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 24123 Lemma for btwnconn1 24132. 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 24124 Lemma for btwnconn1 24132. 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 24125 Lemma for btwnconn1 24132. 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 24126 Lemma for btwnconn1 24132. 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 24127 Lemma for btwnconn1 24132. 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 24128 Lemma for btwnconn1 24132. 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 24129 Lemma for btwnconn1 24132. Using a long string of invocations of linecgr 24112, 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 24130 Lemma for btwnconn1 24132. 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 24131 Lemma for btwnconn1 24132. 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 24132 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 24133 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 24134 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 24135 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 24136* Generalization of axsegcon 23963. 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 23963, 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 >. ) )
 
18.7.37  Segment less than or equal to
 
Syntaxcsegle 24137 Declare the constant for the segment less than or equal to relationship.
 class  Seg<_
 
Definitiondf-segle 24138* 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 24139* 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 24140* 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 24141 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 24142 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 24143 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 24144 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 24145 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 24146 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 24147 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 24148 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 24149 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 >. ) ) ) )
 
18.7.38  Outside of relationship
 
Syntaxcoutsideof 24150 Declare the syntax for the outside of constant.
 class OutsideOf
 
Definitiondf-outsideof 24151 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 24152 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 24153 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 24154 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 24155 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 24156 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 24157* 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 24158 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 24159 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 24160 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 24161 Uniqueness law for OutsideOf. Analog of segconeq 24041. (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 24162* 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 24163 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 24164 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 >. )
 
18.7.39  Lines and Rays
 
Syntaxcline2 24165 Declare the constant for the line function.
 class Line
 
Syntaxcray 24166 Declare the constant for the ray function.
 class Ray
 
Syntaxclines2 24167 Declare the constant for the set of all lines.
 class LinesEE
 
Definitiondf-line2 24168* 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 24169* 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 24170 Define the set of all lines. Definition 6.14, part 2 of [Schwabhauser] p. 45. See ellines 24183 for membership. (Contributed by Scott Fenton, 28-Oct-2013.)
 |- LinesEE  =  ran Line
 
Theoremfunray 24171 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 24172* 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 24173 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 24174 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 24175* 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 24176 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 24177* 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 24178 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 24179 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 24180 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 24181 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 ) )
 
Theoremlinerflx2 24182 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 ) )  ->  Q  e.  ( PLine Q ) )
 
Theoremellines 24183* Membership in the set of all lines. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( A  e. LinesEE  <->  E. n  e.  NN  E. p  e.  ( EE
 `  n ) E. q  e.  ( EE `  n ) ( p  =/=  q  /\  A  =  ( pLine q ) ) )
 
Theoremlinethru 24184 If  A is a line containing two distinct points  P and  Q, then  A is the line through  P and  Q. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e. LinesEE  /\  ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q ) 
 ->  A  =  ( PLine
 Q ) )
 
Theoremhilbert1.1 24185* There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  E. x  e. LinesEE  ( P  e.  x  /\  Q  e.  x ) )
 
Theoremhilbert1.2 24186* There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by NM, 17-Jun-2017.)
 |-  ( P  =/=  Q  ->  E* x  e. LinesEE ( P  e.  x  /\  Q  e.  x ) )
 
Theoremlinethrueu 24187* There is a unique line going through any two distinct points. Theorem 6.19 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  E! x  e. LinesEE  ( P  e.  x  /\  Q  e.  x ) )
 
Theoremlineintmo 24188* Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e. LinesEE  /\  B  e. LinesEE 
 /\  A  =/=  B )  ->  E* x ( x  e.  A  /\  x  e.  B ) )
 
18.7.40  Bernoulli polynomials and sums of k-th powers
 
Syntaxcbp 24189 Declare the constant for the Bernoulli polynomial operator.
 class BernPoly
 
Definitiondf-bpoly 24190* Define the Bernoulli polynomials. Here we use well-founded recursion to define the Bernoulli polynomials. This agrees with most textbook definitions, although explicit formulae do exist. (Contributed by Scott Fenton, 22-May-2014.)
 |- BernPoly  =  ( m  e.  NN0 ,  x  e.  CC  |->  ( U. { f  |  E. s
 ( f  Fn  s  /\  ( s  C_  NN0  /\  A. e  e.  s  Pred (  <  ,  NN0 ,  e )  C_  s ) 
 /\  A. e  e.  s  ( f `  e
 )  =  ( ( g  e.  _V  |->  [_ ( # `  dom  g
 )  /  n ]_ (
 ( x ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
 ( n  -  k
 )  +  1 ) ) ) ) ) `
  ( f  |`  Pred
 (  <  ,  NN0 ,  e ) ) ) ) } `  m ) )
 
Theorembpolylem 24191* Lemma for bpolyval 24192. (Contributed by Scott Fenton, 22-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  G  =  ( g  e.  _V  |->  [_ ( # `  dom  g )  /  n ]_ ( ( X ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `
  k )  /  ( ( n  -  k )  +  1
 ) ) ) ) )   &    |-  F  =  U. { f  |  E. s
 ( f  Fn  s  /\  ( s  C_  NN0  /\  A. e  e.  s  Pred (  <  ,  NN0 ,  e )  C_  s ) 
 /\  A. e  e.  s  ( f `  e
 )  =  ( G `
  ( f  |`  Pred
 (  <  ,  NN0 ,  e ) ) ) ) }   =>    |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
 ( k BernPoly  X )  /  ( ( N  -  k )  +  1
 ) ) ) ) )
 
Theorembpolyval 24192* The value of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.)
 |-  (
 ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
 ( k BernPoly  X )  /  ( ( N  -  k )  +  1
 ) ) ) ) )
 
Theorembpoly0 24193 The value of the Bernoulli polynomials at zero. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( X  e.  CC  ->  ( 0 BernPoly  X )  =  1 )
 
Theorembpoly1 24194 The value of the Bernoulli polynomials at one. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( X  e.  CC  ->  ( 1 BernPoly  X )  =  ( X  -  ( 1 
 /  2 ) ) )
 
Theorembpolycl 24195 Closure law for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
 |-  (
 ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  e.  CC )
 
Theorembpolysum 24196* A sum for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
 |-  (
 ( N  e.  NN0  /\  X  e.  CC )  -> 
 sum_ k  e.  (
 0 ... N ) ( ( N  _C  k
 )  x.  ( ( k BernPoly  X )  /  (
 ( N  -  k
 )  +  1 ) ) )  =  ( X ^ N ) )
 
Theorembpolydiflem 24197* Lemma for bpolydif 24198. (Contributed by Scott Fenton, 12-Jun-2014.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( 1
 ... ( N  -  1 ) ) ) 
 ->  ( ( k BernPoly  ( X  +  1 )
 )  -  ( k BernPoly  X ) )  =  ( k  x.  ( X ^ ( k  -  1 ) ) ) )   =>    |-  ( ph  ->  (
 ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) )  =  ( N  x.  ( X ^ ( N  -  1 ) ) ) )
 
Theorembpolydif 24198 Calculate the difference between successive values of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 26-May-2014.)
 |-  (
 ( N  e.  NN  /\  X  e.  CC )  ->  ( ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) )  =  ( N  x.  ( X ^
 ( N  -  1
 ) ) ) )
 
Theoremfsumkthpow 24199* A closed-form expression for the sum of  K-th powers. (Contributed by Scott Fenton, 16-May-2014.) (Revised by Mario Carneiro, 16-Jun-2014.)
 |-  (
 ( K  e.  NN0  /\  M  e.  NN0 )  -> 
 sum_ n  e.  (
 0 ... M ) ( n ^ K )  =  ( ( ( ( K  +  1 ) BernPoly  ( M  +  1 ) )  -  (
 ( K  +  1 ) BernPoly  0 ) ) 
 /  ( K  +  1 ) ) )
 
Theorembpoly2 24200 The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  ( X  e.  CC  ->  ( 2 BernPoly  X )  =  ( ( ( X ^
 2 )  -  X )  +  ( 1  /  6 ) ) )
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