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Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
18.7.40  Segment less than or equal to
 
Syntaxcsegle 24801 Declare the constant for the segment less than or equal to relationship.
 class  Seg<_
 
Definitiondf-segle 24802* 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 24803* 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 24804* 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 24805 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 24806 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 24807 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 24808 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 24809 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 24810 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 24811 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 24812 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 24813 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 >. ) ) ) )
 
18.7.41  Outside of relationship
 
Syntaxcoutsideof 24814 Declare the syntax for the outside of constant.
 class OutsideOf
 
Definitiondf-outsideof 24815 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 24816 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 24817 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 24818 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 24819 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 24820 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 24821* 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 24822 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 24823 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 24824 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 24825 Uniqueness law for OutsideOf. Analog of segconeq 24705. (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 24826* 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 24827 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 24828 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 >. )
 
18.7.42  Lines and Rays
 
Syntaxcline2 24829 Declare the constant for the line function.
 class Line
 
Syntaxcray 24830 Declare the constant for the ray function.
 class Ray
 
Syntaxclines2 24831 Declare the constant for the set of all lines.
 class LinesEE
 
Definitiondf-line2 24832* 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 24833* 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 24834 Define the set of all lines. Definition 6.14, part 2 of [Schwabhauser] p. 45. See ellines 24847 for membership. (Contributed by Scott Fenton, 28-Oct-2013.)
 |- LinesEE  =  ran Line
 
Theoremfunray 24835 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 24836* 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 24837 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 24838 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 24839* 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 24840 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 24841* 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 24842 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 24843 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 24844 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 24845 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 24846 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 24847* 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 24848 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 24849* 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 24850* 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 24851* 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 24852* 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 ) )
 
18.7.43  Bernoulli polynomials and sums of k-th powers
 
Syntaxcbp 24853 Declare the constant for the Bernoulli polynomial operator.
 class BernPoly
 
Definitiondf-bpoly 24854* 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 24855* Lemma for bpolyval 24856. (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 24856* 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 24857 The value of the Bernoulli polynomials at zero. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( X  e.  CC  ->  ( 0 BernPoly  X )  =  1 )
 
Theorembpoly1 24858 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 24859 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 24860* 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 24861* Lemma for bpolydif 24862. (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 24862 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 24863* 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 24864 The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  ( X  e.  CC  ->  ( 2 BernPoly  X )  =  ( ( ( X ^
 2 )  -  X )  +  ( 1  /  6 ) ) )
 
Theorembpoly3 24865 The Bernoulli polynomials at three. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  ( X  e.  CC  ->  ( 3 BernPoly  X )  =  ( ( ( X ^
 3 )  -  (
 ( 3  /  2
 )  x.  ( X ^ 2 ) ) )  +  ( ( 1  /  2 )  x.  X ) ) )
 
Theorembpoly4 24866 The Bernoulli polynomials at four. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  ( X  e.  CC  ->  ( 4 BernPoly  X )  =  ( ( ( ( X ^ 4 )  -  ( 2  x.  ( X ^ 3 ) ) )  +  ( X ^ 2 ) )  -  ( 1  / ; 3 0 ) ) )
 
Theoremfsumcube 24867* Express the sum of cubes in closed terms. (Contributed by Scott Fenton, 16-Jun-2015.)
 |-  ( T  e.  NN0  ->  sum_ k  e.  ( 0 ... T ) ( k ^
 3 )  =  ( ( ( T ^
 2 )  x.  (
 ( T  +  1 ) ^ 2 ) )  /  4 ) )
 
18.7.44  Rank theorems
 
Theoremrankung 24868 The rank of the union of two sets. Closed form of rankun 7544. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( rank `  ( A  u.  B ) )  =  ( ( rank `  A )  u.  ( rank `  B ) ) )
 
Theoremranksng 24869 The rank of a singleton. Closed form of ranksn 7542. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  ( A  e.  V  ->  (
 rank `  { A }
 )  =  suc  ( rank `  A ) )
 
Theoremrankelg 24870 The membership relation is inherited by the rank function. Closed form of rankel 7527. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  (
 ( B  e.  V  /\  A  e.  B ) 
 ->  ( rank `  A )  e.  ( rank `  B )
 )
 
Theoremrankpwg 24871 The rank of a power set. Closed form of rankpw 7531. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  ( A  e.  V  ->  (
 rank `  ~P A )  =  suc  ( rank `  A ) )
 
Theoremrank0 24872 The rank of the empty set is 
(/) (Contributed by Scott Fenton, 17-Jul-2015.)
 |-  ( rank `  (/) )  =  (/)
 
Theoremrankeq1o 24873 The only set with rank  1o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)
 |-  (
 ( rank `  A )  =  1o  <->  A  =  { (/)
 } )
 
18.7.45  Hereditarily Finite Sets
 
Syntaxchf 24874 The constant Hf is a class.
 class Hf
 
Definitiondf-hf 24875 Define the hereditarily finite sets. These are the finite sets whose elements are finite, and so forth. (Contributed by Scott Fenton, 9-Jul-2015.)
 |- Hf  =  U. ( R1 " om )
 
Theoremelhf 24876* Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.)
 |-  ( A  e. Hf  <->  E. x  e.  om  A  e.  ( R1 `  x ) )
 
Theoremelhf2 24877 Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)
 |-  A  e.  _V   =>    |-  ( A  e. Hf  <->  ( rank `  A )  e.  om )
 
Theoremelhf2g 24878 Hereditarily finiteness via rank. Closed form of elhf2 24877. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  ( A  e.  V  ->  ( A  e. Hf  <->  ( rank `  A )  e.  om )
 )
 
Theorem0hf 24879 The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.)
 |-  (/)  e. Hf
 
Theoremhfun 24880 The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  (
 ( A  e. Hf  /\  B  e. Hf  )  ->  ( A  u.  B )  e. Hf  )
 
Theoremhfsn 24881 The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  ( A  e. Hf  ->  { A }  e. Hf  )
 
Theoremhfadj 24882 Adjoining one HF element to an HF set preserves HF status. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  (
 ( A  e. Hf  /\  B  e. Hf  )  ->  ( A  u.  { B } )  e. Hf  )
 
Theoremhfelhf 24883 Any member of an HF set is itself an HF set. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  (
 ( A  e.  B  /\  B  e. Hf  )  ->  A  e. Hf  )
 
Theoremhftr 24884 The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  Tr Hf
 
Theoremhfext 24885* Extensionality for HF sets depends only on comparison of HF elements. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  (
 ( A  e. Hf  /\  B  e. Hf  )  ->  ( A  =  B  <->  A. x  e. Hf  ( x  e.  A  <->  x  e.  B ) ) )
 
Theoremhfuni 24886 The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  ( A  e. Hf  ->  U. A  e. Hf  )
 
Theoremhfpw 24887 The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  ( A  e. Hf  ->  ~P A  e. Hf  )
 
Theoremhfninf 24888  om is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  -.  om  e. Hf
 
18.8  Mathbox for Anthony Hart
 
18.8.1  Propositional Calculus
 
Theoremtb-ax1 24889 The first of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
Theoremtb-ax2 24890 The second of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Theoremtb-ax3 24891 The third of three axioms in the Tarski-Bernays axiom system.

This axiom, along with ax-mp 8, tb-ax1 24889, and tb-ax2 24890, can be used to derive any theorem or rule that uses only  ->. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  (
 ( ( ph  ->  ps )  ->  ph )  ->  ph )
 
Theoremtbsyl 24892 The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremre1ax2lem 24893 Lemma for re1ax2 24894. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
 
Theoremre1ax2 24894 ax-2 6 rederived from the Tarski-Bernays axiom system. Often tb-ax1 24889 is replaced with this theorem to make a "standard" system. This is because this theorem is easier to work with, despite it being longer. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
 
Theoremnaim1 24895 Constructor theorem for  -/\. (Contributed by Anthony Hart, 1-Sep-2011.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ps  -/\  ch )  ->  ( ph  -/\  ch )
 ) )
 
Theoremnaim2 24896 Constructor theorem for  -/\. (Contributed by Anthony Hart, 1-Sep-2011.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ch  -/\  ps )  ->  ( ch  -/\  ph )
 ) )
 
Theoremnaim1i 24897 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ps  -/\  ch )   =>    |-  ( ph  -/\  ch )
 
Theoremnaim2i 24898 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ch  -/\  ps )   =>    |-  ( ch  -/\  ph )
 
Theoremnaim12i 24899 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ps  -/\  th )   =>    |-  ( ph  -/\  ch )
 
Theoremnabi1 24900 Constructor theorem for  -/\. (Contributed by Anthony Hart, 1-Sep-2011.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ph  -/\  ch )  <->  ( ps  -/\  ch )
 ) )
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