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Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxcontlem3 24001* Lemma for axcont 24011. Given the separation assumption,  B is a subset of  D. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   =>    |-  ( ( ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N ) 
 /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/=  U ) )  ->  B  C_  D )
 
Theoremaxcontlem4 24002* Lemma for axcont 24011. Given the separation assumption,  A is a subset of  D. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   =>    |-  ( ( ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N ) 
 /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  (
 ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  B  =/=  (/) )  /\  Z  =/=  U ) )  ->  A  C_  D )
 
Theoremaxcontlem5 24003* Lemma for axcont 24011. Compute the value of  F. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) ) 
 /\  Z  =/=  U )  /\  P  e.  D )  ->  ( ( F `
  P )  =  T  <->  ( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  (
 ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i ) ) ) ) ) )
 
Theoremaxcontlem6 24004* Lemma for axcont 24011. State the defining properties of the value of  F (Contributed by Scott Fenton, 19-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) ) 
 /\  Z  =/=  U )  /\  P  e.  D )  ->  ( ( F `
  P )  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  (
 ( ( 1  -  ( F `  P ) )  x.  ( Z `
  i ) )  +  ( ( F `
  P )  x.  ( U `  i
 ) ) ) ) )
 
Theoremaxcontlem7 24005* Lemma for axcont 24011. Given two points in  D, one preceeds the other iff its scaling constant is less than the other point's. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) ) 
 /\  Z  =/=  U )  /\  ( P  e.  D  /\  Q  e.  D ) )  ->  ( P 
 Btwn  <. Z ,  Q >.  <-> 
 ( F `  P )  <_  ( F `  Q ) ) )
 
Theoremaxcontlem8 24006* Lemma for axcont 24011. A point in  D is between two others if its function value falls in the middle. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) ) 
 /\  Z  =/=  U )  /\  ( P  e.  D  /\  Q  e.  D  /\  R  e.  D ) )  ->  ( (
 ( F `  P )  <_  ( F `  Q )  /\  ( F `
  Q )  <_  ( F `  R ) )  ->  Q  Btwn  <. P ,  R >. ) )
 
Theoremaxcontlem9 24007* Lemma for axcont 24011. Given the separation assumption, all values of  F over  A are less than or equal to all values of  F over  B. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( N  e.  NN  /\  ( A  C_  ( EE `  N ) 
 /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  (
 ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  B  =/=  (/) )  /\  Z  =/=  U ) )  ->  A. n  e.  ( F " A ) A. m  e.  ( F " B ) n  <_  m )
 
Theoremaxcontlem10 24008* Lemma for axcont 24011. Given a handful of assumptions, derive the conclusion of the final theorem. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( N  e.  NN  /\  ( A  C_  ( EE `  N ) 
 /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  (
 ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  B  =/=  (/) )  /\  Z  =/=  U ) )  ->  E. b  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  b  Btwn  <. x ,  y >. )
 
Theoremaxcontlem11 24009* Lemma for axcont 24011. Eliminate the hypotheses from axcontlem10 24008. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( A  C_  ( EE `  N ) 
 /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  (
 ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  B  =/=  (/) )  /\  Z  =/=  U ) )  ->  E. b  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  b  Btwn  <. x ,  y >. )
 
Theoremaxcontlem12 24010* Lemma for axcont 24011. Eliminate the trivial cases from the previous lemmas. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( A  C_  ( EE `  N ) 
 /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  Z  e.  ( EE `  N ) )  ->  E. b  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  b 
 Btwn  <. x ,  y >. )
 
Theoremaxcont 24011* The axiom of continuity. Take two sets of points  A and 
B. If all the points in  A come before the points of  B on a line, then there is a point separating the two. Axiom A11 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N )  /\  E. a  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  x  Btwn  <. a ,  y >. ) )  ->  E. b  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  b 
 Btwn  <. x ,  y >. )
 
18.7.32  Congruence properties
 
Syntaxcofs 24012 Declare the syntax for the outer five segment configuration.
 class  OuterFiveSeg
 
Definitiondf-ofs 24013* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 23973). See brofs 24035 and 5segofs 24036 for how it is used. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.)
 |-  OuterFiveSeg  =  { <. 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 ) E. x  e.  ( EE `  n ) E. y  e.  ( EE `  n ) E. z  e.  ( EE `  n ) E. w  e.  ( EE `  n ) ( p  =  <. <. a ,  b >. ,  <. c ,  d >. >.  /\  q  =  <.
 <. x ,  y >. , 
 <. z ,  w >. >.  /\  ( ( b  Btwn  <.
 a ,  c >.  /\  y  Btwn  <. x ,  z >. )  /\  ( <. a ,  b >.Cgr <. x ,  y >.  /\ 
 <. b ,  c >.Cgr <.
 y ,  z >. ) 
 /\  ( <. a ,  d >.Cgr <. x ,  w >.  /\  <. b ,  d >.Cgr
 <. y ,  w >. ) ) ) }
 
Theoremcgrrflx2d 24014 Deduction form of axcgrrflx 23949. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   =>    |-  ( ph  ->  <. A ,  B >.Cgr <. B ,  A >. )
 
Theoremcgrtr4d 24015 Deduction form of axcgrtr 23950. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ph  ->  <. A ,  B >.Cgr <. C ,  D >. )   &    |-  ( ph  ->  <. A ,  B >.Cgr <. E ,  F >. )   =>    |-  ( ph  ->  <. C ,  D >.Cgr <. E ,  F >. )
 
Theoremcgrtr4and 24016 Deduction form of axcgrtr 23950. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. C ,  D >. )   &    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. E ,  F >. )   =>    |-  ( ( ph  /\  ps )  ->  <. C ,  D >.Cgr
 <. E ,  F >. )
 
Theoremcgrrflx 24017 Reflexivity law for congruence. Theorem 2.1 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.Cgr <. A ,  B >. )
 
Theoremcgrrflxd 24018 Deduction form of cgrrflx 24017. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   =>    |-  ( ph  ->  <. A ,  B >.Cgr <. A ,  B >. )
 
Theoremcgrcomim 24019 Congruence commutes on the two sides. Implication version. Theorem 2.2 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  ->  <. C ,  D >.Cgr <. A ,  B >. ) )
 
Theoremcgrcom 24020 Congruence commutes between the two sides. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  <->  <. C ,  D >.Cgr <. A ,  B >. ) )
 
Theoremcgrcomand 24021 Deduction form of cgrcom 24020. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. C ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  <. C ,  D >.Cgr
 <. A ,  B >. )
 
Theoremcgrtr 24022 Transitivity law for congruence. Theorem 2.3 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 24-Sep-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 >.Cgr <. C ,  D >.  /\ 
 <. C ,  D >.Cgr <. E ,  F >. ) 
 ->  <. A ,  B >.Cgr
 <. E ,  F >. ) )
 
Theoremcgrtrand 24023 Deduction form of cgrtr 24022. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. C ,  D >. )   &    |-  ( ( ph  /\  ps )  ->  <. C ,  D >.Cgr
 <. E ,  F >. )   =>    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. E ,  F >. )
 
Theoremcgrtr3 24024 Transitivity law for congruence. (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 )  /\  E  e.  ( EE `  N ) 
 /\  F  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
 <. C ,  D >.Cgr <. E ,  F >. ) 
 ->  <. A ,  B >.Cgr
 <. C ,  D >. ) )
 
Theoremcgrtr3and 24025 Deduction form of cgrtr3 24024. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. E ,  F >. )   &    |-  ( ( ph  /\  ps )  ->  <. C ,  D >.Cgr
 <. E ,  F >. )   =>    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. C ,  D >. )
 
Theoremcgrcoml 24026 Congruence commutes on the left. Biconditional version of Theorem 2.4 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  <->  <. B ,  A >.Cgr <. C ,  D >. ) )
 
Theoremcgrcomr 24027 Congruence commutes on the right. Biconditional version of Theorem 2.5 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  <->  <. A ,  B >.Cgr <. D ,  C >. ) )
 
Theoremcgrcomlr 24028 Congruence commutes on both sides. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  <->  <. B ,  A >.Cgr <. D ,  C >. ) )
 
Theoremcgrcomland 24029 Deduction form of cgrcoml 24026. (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  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. C ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  <. B ,  A >.Cgr
 <. C ,  D >. )
 
Theoremcgrcomrand 24030 Deduction form of cgrcoml 24026. (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  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. C ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. D ,  C >. )
 
Theoremcgrcomlrand 24031 Deduction form of cgrcomlr 24028. (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  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. C ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  <. B ,  A >.Cgr
 <. D ,  C >. )
 
Theoremcgrtriv 24032 Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  A >.Cgr <. B ,  B >. )
 
Theoremcgrid2 24033 Identity law for congruence. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( <. A ,  A >.Cgr
 <. B ,  C >.  ->  B  =  C )
 )
 
Theoremcgrdegen 24034 Two congruent segments are either both degenrate or both non-degenerate. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  ->  ( A  =  B  <->  C  =  D ) ) )
 
Theorembrofs 24035 Binary relationship form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-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 >.  /\  F  Btwn  <. E ,  G >. ) 
 /\  ( <. A ,  B >.Cgr <. E ,  F >.  /\  <. B ,  C >.Cgr
 <. F ,  G >. ) 
 /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr
 <. F ,  H >. ) ) ) )
 
Theorem5segofs 24036 Rephrase ax5seg 23973 using the outer five segment predicate. Theorem 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-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 >. >.  /\  A  =/=  B ) 
 ->  <. C ,  D >.Cgr
 <. G ,  H >. ) )
 
Theoremofscom 24037 The outer five segment predicate commutes. (Contributed by Scott Fenton, 26-Sep-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 >. >.  <->  <. <. E ,  F >. , 
 <. G ,  H >. >.  OuterFiveSeg  <. <. A ,  B >. , 
 <. C ,  D >. >.
 ) )
 
Theoremcgrextend 24038 Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Scott Fenton, 12-Jun-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 >.  /\  E  Btwn  <. D ,  F >. ) 
 /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
 <. E ,  F >. ) )  ->  <. A ,  C >.Cgr <. D ,  F >. ) )
 
Theoremcgrextendand 24039 Deduction form of cgrextend 24038. (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  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  B  Btwn  <. A ,  C >. )   &    |-  ( ( ph  /\ 
 ps )  ->  E  Btwn  <. D ,  F >. )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. D ,  E >. )   &    |-  ( ( ph  /\ 
 ps )  ->  <. B ,  C >.Cgr <. E ,  F >. )   =>    |-  ( ( ph  /\  ps )  ->  <. A ,  C >.Cgr
 <. D ,  F >. )
 
Theoremsegconeq 24040 Two points that satsify the conclusion of axsegcon 23962 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N ) 
 /\  Y  e.  ( EE `  N ) ) )  ->  ( ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
 <. B ,  C >. ) 
 /\  ( A  Btwn  <. Q ,  Y >.  /\ 
 <. A ,  Y >.Cgr <. B ,  C >. ) )  ->  X  =  Y ) )
 
Theoremsegconeu 24041* Existential uniqueness version of segconeq 24040. (Contributed by Scott Fenton, 19-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 ) ) 
 /\  C  =/=  D ) )  ->  E! r  e.  ( EE `  N ) ( D  Btwn  <. C ,  r >.  /\ 
 <. D ,  r >.Cgr <. A ,  B >. ) )
 
18.7.33  Betweenness properties
 
Theorembtwntriv2 24042 Betweeness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  B  Btwn  <. A ,  B >. )
 
Theorembtwncomim 24043 Betweeness commutes. Implication version. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Btwn  <. B ,  C >.  ->  A  Btwn  <. C ,  B >. ) )
 
Theorembtwncom 24044 Betweeness commutes. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  Btwn  <. B ,  C >. 
 <->  A  Btwn  <. C ,  B >. ) )
 
Theorembtwncomand 24045 Deduction form of btwncom 24044. (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  /\  ps )  ->  A  Btwn  <. B ,  C >. )   =>    |-  ( ( ph  /\  ps )  ->  A  Btwn  <. C ,  B >. )
 
Theorembtwntriv1 24046 Betweeness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  A  Btwn  <. A ,  B >. )
 
Theorembtwnswapid 24047 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 Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( ( A  Btwn  <. B ,  C >.  /\  B  Btwn  <. A ,  C >. )  ->  A  =  B ) )
 
Theorembtwnswapid2 24048 If you can swap arguments one and three of a betweenness statement, then those arguments are identical. (Contributed by Scott Fenton, 7-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( ( A  Btwn  <. B ,  C >.  /\  C  Btwn  <. B ,  A >. )  ->  A  =  C ) )
 
Theorembtwnintr 24049 Inner transitivity law for betweenness. Left hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-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  <. B ,  D >. )  ->  B  Btwn  <. A ,  C >. ) )
 
Theorembtwnexch3 24050 Exchange the first endpoint in betweenness. Left hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  D >. )  ->  C  Btwn  <. B ,  D >. ) )
 
Theorembtwnexch3and 24051 Deduction form of btwnexch3 24050. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  B  Btwn  <. A ,  C >. )   &    |-  ( ( ph  /\ 
 ps )  ->  C  Btwn  <. A ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  C  Btwn  <. B ,  D >. )
 
Theorembtwnouttr2 24052 Outer transitivity law for betweenness. Left hand side of Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( B  =/=  C 
 /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) 
 ->  C  Btwn  <. A ,  D >. ) )
 
Theorembtwnexch2 24053 Exchange the outer point of two betweenness statements. Right hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 14-Jun-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  <. B ,  D >. )  ->  C  Btwn  <. A ,  D >. ) )
 
Theorembtwnouttr 24054 Outer transitivity law for betweenness. Right hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 14-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( B  =/=  C 
 /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) 
 ->  B  Btwn  <. A ,  D >. ) )
 
Theorembtwnexch 24055 Outer transitivity law for betweenness. Right hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 24-Sep-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( ( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  D >. )  ->  B  Btwn  <. A ,  D >. ) )
 
Theorembtwnexchand 24056 Deduction form of btwnexch 24055. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  B  Btwn  <. A ,  C >. )   &    |-  ( ( ph  /\ 
 ps )  ->  C  Btwn  <. A ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  B  Btwn  <. A ,  D >. )
 
Theorembtwndiff 24057* There is always a  c distinct from  B such that  B lies between  A and  c. Theorem 3.14 of [Schwabhauser] p. 32. (Contributed by Scott Fenton, 24-Sep-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  E. c  e.  ( EE `  N ) ( B  Btwn  <. A ,  c >.  /\  B  =/=  c ) )
 
Theoremtrisegint 24058* 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 Scott Fenton, 24-Sep-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) 
 /\  P  e.  ( EE `  N ) ) )  ->  ( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) 
 ->  E. q  e.  ( EE `  N ) ( q  Btwn  <. P ,  C >.  /\  q  Btwn  <. B ,  E >. ) ) )
 
18.7.34  Segment Transportation
 
Syntaxctransport 24059 Declare the syntax for the segment transport function.
 class TransportTo
 
Definitiondf-transport 24060* Define the segment transport function. See fvtransport 24062 for an explanation of the function. (Contributed by Scott Fenton, 18-Oct-2013.)
 |- TransportTo  =  { <.
 <. p ,  q >. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( ( EE
 `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n )  X.  ( EE `  n ) ) 
 /\  ( 1st `  q
 )  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  /\  <. ( 2nd `  q
 ) ,  r >.Cgr p ) ) ) }
 
Theoremfuntransport 24061 The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun TransportTo
 
Theoremfvtransport 24062* Calculate the value of the TransportTo function. This function takes four points,  A through  D, where  C and  D are distinct. It then returns the point that extends  C D by the length of  A B. (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 ) 
 /\  D  e.  ( EE `  N ) ) 
 /\  C  =/=  D ) )  ->  ( <. A ,  B >.TransportTo <. C ,  D >. )  =  (
 iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C ,  r >.  /\  <. D ,  r >.Cgr <. A ,  B >. ) ) )
 
Theoremtransportcl 24063 Closure law for segment transport. (Contributed by Scott Fenton, 19-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 ) ) 
 /\  C  =/=  D ) )  ->  ( <. A ,  B >.TransportTo <. C ,  D >. )  e.  ( EE `  N ) )
 
Theoremtransportprops 24064 Calculate the defining properties of the transport function (Contributed by Scott Fenton, 19-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 ) ) 
 /\  C  =/=  D ) )  ->  ( D 
 Btwn  <. C ,  ( <. A ,  B >.TransportTo <. C ,  D >. ) >.  /\ 
 <. D ,  ( <. A ,  B >.TransportTo <. C ,  D >. ) >.Cgr <. A ,  B >. ) )
 
18.7.35  Properties relating betweenness and congruence
 
Syntaxcifs 24065 Declare the syntax for the inner five segment predicate.
 class  InnerFiveSeg
 
Syntaxccgr3 24066 Declare the syntax for the three place congruence predicate.
 class Cgr3
 
Syntaxccolin 24067 Declare the syntax for the colinearity predicate.
 class  Colinear
 
Syntaxcfs 24068 Declare the syntax for the five segment predicate.
 class  FiveSeg
 
Definitiondf-ifs 24069* The inner five segment configuration is an abbreviation for another congruence condition. See brifs 24073 and ifscgr 24074 for how it is used. Definition 4.1 of [Schwabhauser] p. 34. (Contributed by Scott Fenton, 26-Sep-2013.)
 |-  InnerFiveSeg  =  { <. 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 ) E. x  e.  ( EE `  n ) E. y  e.  ( EE `  n ) E. z  e.  ( EE `  n ) E. w  e.  ( EE `  n ) ( p  =  <. <. a ,  b >. ,  <. c ,  d >. >.  /\  q  =  <.
 <. x ,  y >. , 
 <. z ,  w >. >.  /\  ( ( b  Btwn  <.
 a ,  c >.  /\  y  Btwn  <. x ,  z >. )  /\  ( <. a ,  c >.Cgr <. x ,  z >.  /\ 
 <. b ,  c >.Cgr <.
 y ,  z >. ) 
 /\  ( <. a ,  d >.Cgr <. x ,  w >.  /\  <. c ,  d >.Cgr
 <. z ,  w >. ) ) ) }
 
Definitiondf-cgr3 24070* The three place congruence predicate. This is an abbreviation for saying that all three pair in a triple are congruent with each other. Three place form of Definition 4.4 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
 |- Cgr3  =  { <. 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 ) E. e  e.  ( EE `  n ) E. f  e.  ( EE `  n ) ( p  = 
 <. a ,  <. b ,  c >. >.  /\  q  =  <. d ,  <. e ,  f >. >.  /\  ( <. a ,  b >.Cgr <. d ,  e >.  /\  <. a ,  c >.Cgr <. d ,  f >.  /\  <. b ,  c >.Cgr
 <. e ,  f >. ) ) }
 
Definitiondf-colinear 24071* The colinearity predicate states that the three points in its arguments sit on one line. Definition 4.10 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 25-Oct-2013.)
 |-  Colinear  =  `' { <. <. b ,  c >. ,  a >.  |  E. n  e.  NN  (
 ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n ) )  /\  ( a 
 Btwn  <. b ,  c >.  \/  b  Btwn  <. c ,  a >.  \/  c  Btwn  <. a ,  b >. ) ) }
 
Definitiondf-fs 24072* The general five segment configuration is a generalization of the outer and inner five segment configurations. See brfs 24109 and fscgr 24110 for its use. Definition 4.15 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
 |-  FiveSeg  =  { <. 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 ) E. x  e.  ( EE `  n ) E. y  e.  ( EE `  n ) E. z  e.  ( EE `  n ) E. w  e.  ( EE `  n ) ( p  =  <. <. a ,  b >. ,  <. c ,  d >.
 >.  /\  q  =  <. <. x ,  y >. , 
 <. z ,  w >. >.  /\  ( a  Colinear  <. b ,  c >.  /\  <. a , 
 <. b ,  c >. >.Cgr3 <. x ,  <. y ,  z >. >.  /\  ( <. a ,  d >.Cgr <. x ,  w >.  /\  <. b ,  d >.Cgr <. y ,  w >. ) ) ) }
 
Theorembrifs 24073 Binary relationship form of the inner five segment predicate. (Contributed by Scott Fenton, 26-Sep-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 >.  /\  F  Btwn  <. E ,  G >. ) 
 /\  ( <. A ,  C >.Cgr <. E ,  G >.  /\  <. B ,  C >.Cgr
 <. F ,  G >. ) 
 /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. C ,  D >.Cgr
 <. G ,  H >. ) ) ) )
 
Theoremifscgr 24074 Inner five segment congruence. Take two triangles,  A D C and  E H G, with 
B between  A and  C and  F between  E and  G. 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 Scott Fenton, 27-Sep-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 ,  D >.Cgr <. F ,  H >. ) )
 
Theoremcgrsub 24075 Removing identical parts from the end of a line segment preserves congruence. Theorem 4.3 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 )  /\  E  e.  ( EE `  N ) 
 /\  F  e.  ( EE `  N ) ) )  ->  ( (
 ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. ) 
 /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
 <. E ,  F >. ) )  ->  <. A ,  B >.Cgr <. D ,  E >. ) )
 
Theorembrcgr3 24076 Binary relationship form of the three-place congruence predicate. (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 ) ) )  ->  ( <. A ,  <. B ,  C >.
 >.Cgr3 <. D ,  <. E ,  F >. >.  <->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr
 <. D ,  F >.  /\ 
 <. B ,  C >.Cgr <. E ,  F >. ) ) )
 
Theoremcgr3permute3 24077 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 >. >.  <->  <. B ,  <. C ,  A >. >.Cgr3 <. E ,  <. F ,  D >. >.
 ) )
 
Theoremcgr3permute1 24078 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 >. >.  <->  <. A ,  <. C ,  B >. >.Cgr3 <. D ,  <. F ,  E >. >.
 ) )
 
Theoremcgr3permute2 24079 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 >. >.  <->  <. B ,  <. A ,  C >. >.Cgr3 <. E ,  <. D ,  F >. >.
 ) )
 
Theoremcgr3permute4 24080 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 ,  <. A ,  B >. >.Cgr3 <. F ,  <. D ,  E >. >.
 ) )
 
Theoremcgr3permute5 24081 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 24082 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 24083 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 24084 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 24085* 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 24086 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 24087 Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Rel  Colinear
 
Theorembrcolinear2 24088* 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 24089 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 24090 The colinear predicate exists. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Colinear  e.  _V
 
Theoremcolineardim1 24091 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 24092 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 24093 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 24094 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 24095 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 24096 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 24097 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 24098 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 24099 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 24100 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 >. ) )
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