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Theorem List for Metamath Proof Explorer - 23801-23900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisperp2d 23801 One direction of isperp2 23800 (Contributed by Thierry Arnoux, 10-Nov-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  ran  L )   &    |-  ( ph  ->  B  e.  ran  L )   &    |-  ( ph  ->  X  e.  ( A  i^i  B ) )   &    |-  ( ph  ->  U  e.  A )   &    |-  ( ph  ->  V  e.  B )   &    |-  ( ph  ->  A (⟂G `  G ) B )   =>    |-  ( ph  ->  <" U X V ">  e.  (∟G `  G ) )
 
Theoremragperp 23802 Deduce that two lines are perpendicular from a right angle statement. One direction of theorem 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 20-Oct-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  ran  L )   &    |-  ( ph  ->  B  e.  ran  L )   &    |-  ( ph  ->  X  e.  ( A  i^i  B ) )   &    |-  ( ph  ->  U  e.  A )   &    |-  ( ph  ->  V  e.  B )   &    |-  ( ph  ->  U  =/=  X )   &    |-  ( ph  ->  V  =/=  X )   &    |-  ( ph  ->  <" U X V ">  e.  (∟G `  G )
 )   =>    |-  ( ph  ->  A (⟂G `
  G ) B )
 
Theoremfootex 23803* Lemma for foot 23804: existence part. (Contributed by Thierry Arnoux, 19-Oct-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  ran  L )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  -.  C  e.  A )   =>    |-  ( ph  ->  E. x  e.  A  ( C L x ) (⟂G `  G ) A )
 
Theoremfoot 23804* From a point  C outside of a line  A, there exists a unique point  x on  A such that  ( C L x ) is perpendicular to  A. That point is called the foot from  C on  A. Theorem 8.18 of [Schwabhauser] p. 60. (Contributed by Thierry Arnoux, 19-Oct-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  ran  L )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  -.  C  e.  A )   =>    |-  ( ph  ->  E! x  e.  A  ( C L x ) (⟂G `  G ) A )
 
Theoremperprag 23805 Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 10-Nov-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  ( A L B ) )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  ( A L B ) (⟂G `  G ) ( C L D ) )   =>    |-  ( ph  ->  <" A C D ">  e.  (∟G `  G ) )
 
TheoremperpdragALT 23806 Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 12-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D (⟂G `  G ) ( B L C ) )   =>    |-  ( ph  ->  <" A B C ">  e.  (∟G `  G )
 )
 
Theoremperpdrag 23807 Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 12-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D (⟂G `  G ) ( B L C ) )   =>    |-  ( ph  ->  <" A B C ">  e.  (∟G `  G )
 )
 
Theoremcolperp 23808 Deduce a perpendicularity from perpendicularity and colinearity. (Contributed by Thierry Arnoux, 8-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  ( A L B ) (⟂G `  G ) D )   &    |-  ( ph  ->  ( C  e.  ( A L B )  \/  A  =  B ) )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  ( A L C ) (⟂G `  G ) D )
 
Theoremcolperpexlem1 23809 Lemma for colperp 23808. First part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 27-Oct-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  S  =  (pInvG `  G )   &    |-  M  =  ( S `  A )   &    |-  N  =  ( S `
  B )   &    |-  K  =  ( S `  Q )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  Q  e.  P )   &    |-  ( ph  ->  <" A B C ">  e.  (∟G `  G )
 )   &    |-  ( ph  ->  ( K `  ( M `  C ) )  =  ( N `  C ) )   =>    |-  ( ph  ->  <" B A Q ">  e.  (∟G `  G ) )
 
Theoremcolperpexlem2 23810 Lemma for colperpex 23812. Second part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 10-Nov-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  S  =  (pInvG `  G )   &    |-  M  =  ( S `  A )   &    |-  N  =  ( S `
  B )   &    |-  K  =  ( S `  Q )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  Q  e.  P )   &    |-  ( ph  ->  <" A B C ">  e.  (∟G `  G )
 )   &    |-  ( ph  ->  ( K `  ( M `  C ) )  =  ( N `  C ) )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  A  =/=  Q )
 
Theoremcolperpexlem3 23811* Lemma for colperpex 23812. Case 1 of theorem 8.21 of [Schwabhauser] p. 63. (Contributed by Thierry Arnoux, 20-Nov-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  -.  C  e.  ( A L B ) )   =>    |-  ( ph  ->  E. p  e.  P  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  E. t  e.  P  ( ( t  e.  ( A L B )  \/  A  =  B ) 
 /\  t  e.  ( C I p ) ) ) )
 
Theoremcolperpex 23812* In dimension 2 and above, on a line 
( A L B ) there is always a perpendicular  P from  A on a given plane (here given by  C, in case  C does not lie on the line). Theorem 8.21 of [Schwabhauser] p. 63. (Contributed by Thierry Arnoux, 20-Nov-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  GDimTarskiG 2 )   =>    |-  ( ph  ->  E. p  e.  P  ( ( A L p ) (⟂G `  G ) ( A L B )  /\  E. t  e.  P  ( ( t  e.  ( A L B )  \/  A  =  B ) 
 /\  t  e.  ( C I p ) ) ) )
 
Theoremmideulem 23813* Lemma for mideu 23814. We can assume mideulem.9 "without loss of generality" (Contributed by Thierry Arnoux, 25-Nov-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  S  =  (pInvG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  Q  e.  P )   &    |-  ( ph  ->  O  e.  P )   &    |-  ( ph  ->  T  e.  P )   &    |-  ( ph  ->  ( A L B ) (⟂G `  G ) ( Q L B ) )   &    |-  ( ph  ->  ( A L B ) (⟂G `  G ) ( A L O ) )   &    |-  ( ph  ->  T  e.  ( A L B ) )   &    |-  ( ph  ->  T  e.  ( Q I O ) )   &    |-  ( ph  ->  ( A  .-  O )
 (≤G `  G )
 ( B  .-  Q ) )   =>    |-  ( ph  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A ) )
 
Theoremmideu 23814* Existence and uniqueness of the midpoint, Theorem 8.22 of [Schwabhauser] p. 64. Note that this proof requires a construction in 2 dimensions or more, i.e. it does not prove the existence of a midpoint in dimension 1, for a geometry restricted to a line. (Contributed by Thierry Arnoux, 25-Nov-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  S  =  (pInvG `  G )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  GDimTarskiG 2 )   =>    |-  ( ph  ->  E! x  e.  P  B  =  ( ( S `  x ) `  A ) )
 
15.2.13  Midpoints and Line Mirroring
 
Syntaxcmid 23815 Declare the constant for the midpoint operation.
 class midG
 
Syntaxclmi 23816 Declare the constant for the line mirroring function.
 class lInvG
 
Definitiondf-mid 23817* Define the midpoint operation. Definition 10.1 of [Schwabhauser] p. 88. (Contributed by Thierry Arnoux, 9-Jun-2019.)
 |- midG  =  ( g  e.  _V  |->  ( a  e.  ( Base `  g ) ,  b  e.  ( Base `  g )  |->  ( iota_ m  e.  ( Base `  g
 ) b  =  ( ( (pInvG `  g
 ) `  m ) `  a ) ) ) )
 
Definitiondf-lmi 23818* Define the line mirroring function. Definition 10.3 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 1-Dec-2019.)
 |- lInvG  =  ( g  e.  _V  |->  ( m  e.  ran  (LineG `  g )  |->  ( a  e.  ( Base `  g )  |->  ( iota_ b  e.  ( Base `  g
 ) ( ( a (midG `  g )
 b )  e.  m  /\  ( m (⟂G `  g
 ) ( a (LineG `  g ) b )  \/  a  =  b ) ) ) ) ) )
 
Theoremmidf 23819 Midpoint as a function. (Contributed by Thierry Arnoux, 1-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   =>    |-  ( ph  ->  (midG `  G ) : ( P  X.  P ) --> P )
 
Theoremmidcl 23820 Closure of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( A (midG `  G ) B )  e.  P )
 
Theoremismidb 23821 Property of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  S  =  (pInvG `  G )   &    |-  ( ph  ->  M  e.  P )   =>    |-  ( ph  ->  ( B  =  ( ( S `  M ) `
  A )  <->  ( A (midG `  G ) B )  =  M ) )
 
Theoremmidbtwn 23822 Betweenness of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( A (midG `  G ) B )  e.  ( A I B ) )
 
Theoremmidcgr 23823 Congruence of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  ( A (midG `  G ) B )  =  C )   =>    |-  ( ph  ->  ( C  .-  A )  =  ( C  .-  B ) )
 
Theoremmidid 23824 Midpoint of a null segment. (Contributed by Thierry Arnoux, 7-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( A (midG `  G ) A )  =  A )
 
Theoremmidcom 23825 Commutativity rule for the midpoint. (Contributed by Thierry Arnoux, 2-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( A (midG `  G ) B )  =  ( B (midG `  G ) A ) )
 
Theoremmirmid 23826 Point inversion preserves midpoints. (Contributed by Thierry Arnoux, 12-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  S  =  ( (pInvG `  G ) `  M )   &    |-  ( ph  ->  M  e.  P )   =>    |-  ( ph  ->  (
 ( S `  A ) (midG `  G )
 ( S `  B ) )  =  ( S `  ( A (midG `  G ) B ) ) )
 
Theoremlmieu 23827* Uniqueness of the line mirror point. Theorem 10.2 of [Schwabhauser] p. 88. (Contributed by Thierry Arnoux, 1-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   &    |-  ( ph  ->  A  e.  P )   =>    |-  ( ph  ->  E! b  e.  P  (
 ( A (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G ) ( A L b )  \/  A  =  b ) ) )
 
Theoremlmif 23828 Line mirror as a function. (Contributed by Thierry Arnoux, 11-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   =>    |-  ( ph  ->  M : P --> P )
 
Theoremlmicl 23829 Closure of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   &    |-  ( ph  ->  A  e.  P )   =>    |-  ( ph  ->  ( M `  A )  e.  P )
 
Theoremislmib 23830 Property of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( B  =  ( M `  A )  <->  ( ( A (midG `  G ) B )  e.  D  /\  ( D (⟂G `  G ) ( A L B )  \/  A  =  B ) ) ) )
 
Theoremlmicom 23831 The line mirroring function is an involution. Theorem 10.4 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  ( M `  A )  =  B )   =>    |-  ( ph  ->  ( M `  B )  =  A )
 
Theoremlmilmi 23832 Line mirroring is an involution. Theorem 10.5 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   &    |-  ( ph  ->  A  e.  P )   =>    |-  ( ph  ->  ( M `  ( M `  A ) )  =  A )
 
Theoremlmireu 23833* Any point has a unique antecedent through line mirroring. Theorem 10.6 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   &    |-  ( ph  ->  A  e.  P )   =>    |-  ( ph  ->  E! b  e.  P  ( M `  b )  =  A )
 
Theoremlmieq 23834 Equality deduction for line mirroring. Theorem 10.7 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  ( M `  A )  =  ( M `  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremlmiinv 23835 The invariants of the line mirroring lie on the mirror line. Theorem 10.8 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   &    |-  ( ph  ->  A  e.  P )   =>    |-  ( ph  ->  (
 ( M `  A )  =  A  <->  A  e.  D ) )
 
Theoremlmimid 23836 If we have a right angle, then the mirror point is the point inversion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   &    |-  ( ph  ->  A  e.  P )   &    |-  S  =  ( (pInvG `  G ) `  B )   &    |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( M `  C )  =  ( S `  C ) )
 
Theoremlmif1o 23837 The line mirroring function  M is a bijection. Theorem 10.9 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   =>    |-  ( ph  ->  M : P -1-1-onto-> P )
 
Theoremlmiisolem 23838 Lemma for lmiiso 23839 (Contributed by Thierry Arnoux, 14-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  S  =  ( (pInvG `  G ) `  Z )   &    |-  Z  =  ( ( A (midG `  G ) ( M `
  A ) ) (midG `  G )
 ( B (midG `  G ) ( M `
  B ) ) )   =>    |-  ( ph  ->  (
 ( M `  A )  .-  ( M `  B ) )  =  ( A  .-  B ) )
 
Theoremlmiiso 23839 The line mirroring function is an isometry, i.e. it is conserves congruence. Because it is also a bijection, it is also a motion. Theorem 10.10 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  (
 ( M `  A )  .-  ( M `  B ) )  =  ( A  .-  B ) )
 
Theoremlmimot 23840 Line mirroring is a motion of the geometric space. Theorem 10.11 of [Schwabhauser] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  M  =  ( (lInvG `  G ) `  D )   &    |-  L  =  (LineG `  G )   &    |-  ( ph  ->  D  e.  ran  L )   =>    |-  ( ph  ->  M  e.  ( GIsmt G ) )
 
Theoremhypcgrlem1 23841 Lemma for hypcgr 23843, case where triangles share a cathetus. (Contributed by Thierry Arnoux, 15-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  E  e.  P )   &    |-  ( ph  ->  F  e.  P )   &    |-  ( ph  ->  <" A B C ">  e.  (∟G `  G )
 )   &    |-  ( ph  ->  <" D E F ">  e.  (∟G `  G ) )   &    |-  ( ph  ->  ( A  .-  B )  =  ( D  .-  E )
 )   &    |-  ( ph  ->  ( B  .-  C )  =  ( E  .-  F ) )   &    |-  ( ph  ->  B  =  E )   &    |-  S  =  ( (lInvG `  G ) `  ( ( A (midG `  G ) D ) (LineG `  G ) B ) )   &    |-  ( ph  ->  C  =  F )   =>    |-  ( ph  ->  ( A  .-  C )  =  ( D  .-  F ) )
 
Theoremhypcgrlem2 23842 Lemma for hypcgr 23843, case where triangles share one vertex  B. (Contributed by Thierry Arnoux, 16-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  E  e.  P )   &    |-  ( ph  ->  F  e.  P )   &    |-  ( ph  ->  <" A B C ">  e.  (∟G `  G )
 )   &    |-  ( ph  ->  <" D E F ">  e.  (∟G `  G ) )   &    |-  ( ph  ->  ( A  .-  B )  =  ( D  .-  E )
 )   &    |-  ( ph  ->  ( B  .-  C )  =  ( E  .-  F ) )   &    |-  ( ph  ->  B  =  E )   &    |-  S  =  ( (lInvG `  G ) `  ( ( C (midG `  G ) F ) (LineG `  G ) B ) )   =>    |-  ( ph  ->  ( A  .-  C )  =  ( D  .-  F ) )
 
Theoremhypcgr 23843 If the catheti of two right-angled triangles are congruent, so is their hypothenuse. Theorem 10.12 of [Schwabhauser] p. 91. (Contributed by Thierry Arnoux, 16-Dec-2019.)
 |-  P  =  ( Base `  G )   &    |-  .-  =  ( dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  GDimTarskiG 2 )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  E  e.  P )   &    |-  ( ph  ->  F  e.  P )   &    |-  ( ph  ->  <" A B C ">  e.  (∟G `  G )
 )   &    |-  ( ph  ->  <" D E F ">  e.  (∟G `  G ) )   &    |-  ( ph  ->  ( A  .-  B )  =  ( D  .-  E )
 )   &    |-  ( ph  ->  ( B  .-  C )  =  ( E  .-  F ) )   =>    |-  ( ph  ->  ( A  .-  C )  =  ( D  .-  F ) )
 
15.2.14  Congruence of angles
 
Syntaxccgra 23844 Declare the constant for the congruence between angles relation.
 class cgrA
 
Definitiondf-cgra 23845* Define the congruence relation bewteen angles. See definition 11.2 of [Schwabhauser] p. 95. As for triangles we use "words of points". (Contributed by Thierry Arnoux, 27-Nov-2019.)
 |- cgrA  =  ( g  e.  _V  |->  {
 <. a ,  b >.  |  ( ( a  e.  ( ( Base `  g
 )  ^m  ( 0..^ 3 ) )  /\  b  e.  ( ( Base `  g )  ^m  ( 0..^ 3 ) ) )  /\  ( ( b `  0 )  =/=  ( b `  1 )  /\  ( b `
  1 )  =/=  ( b `  2
 )  /\  E. x  e.  ( ( a `  0 ) (LineG `  g ) ( a `
  1 ) ) E. y  e.  (
 ( a `  1
 ) (LineG `  g
 ) ( a `  2 ) ) <" x ( a `  1 ) y "> (cgrG `  g )
 b ) ) }
 )
 
15.2.15  Equilateral triangles
 
Syntaxceqlg 23846 Declare the class of equilateral triangles.
 class eqltrG
 
Definitiondf-eqlg 23847* Define the class of equilateral triangles. (Contributed by Thierry Arnoux, 27-Nov-2019.)
 |- eqltrG  =  ( g  e.  _V  |->  { x  e.  ( (
 Base `  g )  ^m  ( 0..^ 3 ) )  |  x (cgrG `  g ) <" ( x `  1 ) ( x `  2 ) ( x `  0
 ) "> } )
 
15.3  Properties of geometries
 
15.3.1  Isomorphisms between geometries
 
Theoremf1otrgds 23848* Convenient lemma for f1otrg 23850 (Contributed by Thierry Arnoux, 19-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  D  =  (
 dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  B  =  ( Base `  H )   &    |-  E  =  ( dist `  H )   &    |-  J  =  (Itv `  H )   &    |-  ( ph  ->  F : B -1-1-onto-> P )   &    |-  ( ( ph  /\  (
 e  e.  B  /\  f  e.  B )
 )  ->  ( e E f )  =  ( ( F `  e ) D ( F `  f ) ) )   &    |-  ( ( ph  /\  ( e  e.  B  /\  f  e.  B  /\  g  e.  B ) )  ->  ( g  e.  ( e J f )  <->  ( F `  g )  e.  (
 ( F `  e
 ) I ( F `
  f ) ) ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X E Y )  =  ( ( F `  X ) D ( F `  Y ) ) )
 
Theoremf1otrgitv 23849* Convenient lemma for f1otrg 23850 (Contributed by Thierry Arnoux, 19-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  D  =  (
 dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  B  =  ( Base `  H )   &    |-  E  =  ( dist `  H )   &    |-  J  =  (Itv `  H )   &    |-  ( ph  ->  F : B -1-1-onto-> P )   &    |-  ( ( ph  /\  (
 e  e.  B  /\  f  e.  B )
 )  ->  ( e E f )  =  ( ( F `  e ) D ( F `  f ) ) )   &    |-  ( ( ph  /\  ( e  e.  B  /\  f  e.  B  /\  g  e.  B ) )  ->  ( g  e.  ( e J f )  <->  ( F `  g )  e.  (
 ( F `  e
 ) I ( F `
  f ) ) ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( Z  e.  ( X J Y )  <->  ( F `  Z )  e.  (
 ( F `  X ) I ( F `  Y ) ) ) )
 
Theoremf1otrg 23850* A bijection between bases which conserves distances and intervals conserves also geometries. (Contributed by Thierry Arnoux, 23-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  D  =  (
 dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  B  =  ( Base `  H )   &    |-  E  =  ( dist `  H )   &    |-  J  =  (Itv `  H )   &    |-  ( ph  ->  F : B -1-1-onto-> P )   &    |-  ( ( ph  /\  (
 e  e.  B  /\  f  e.  B )
 )  ->  ( e E f )  =  ( ( F `  e ) D ( F `  f ) ) )   &    |-  ( ( ph  /\  ( e  e.  B  /\  f  e.  B  /\  g  e.  B ) )  ->  ( g  e.  ( e J f )  <->  ( F `  g )  e.  (
 ( F `  e
 ) I ( F `
  f ) ) ) )   &    |-  ( ph  ->  H  e.  V )   &    |-  ( ph  ->  G  e. TarskiG )   &    |-  ( ph  ->  (LineG `  H )  =  ( x  e.  B ,  y  e.  ( B  \  { x }
 )  |->  { z  e.  B  |  ( z  e.  ( x J y )  \/  x  e.  ( z J y )  \/  y  e.  ( x J z ) ) } ) )   =>    |-  ( ph  ->  H  e. TarskiG )
 
Theoremf1otrge 23851* A bijection between bases which conserves distances and intervals conserves also the property of being a Euclidean geometry. (Contributed by Thierry Arnoux, 23-Mar-2019.)
 |-  P  =  ( Base `  G )   &    |-  D  =  (
 dist `  G )   &    |-  I  =  (Itv `  G )   &    |-  B  =  ( Base `  H )   &    |-  E  =  ( dist `  H )   &    |-  J  =  (Itv `  H )   &    |-  ( ph  ->  F : B -1-1-onto-> P )   &    |-  ( ( ph  /\  (
 e  e.  B  /\  f  e.  B )
 )  ->  ( e E f )  =  ( ( F `  e ) D ( F `  f ) ) )   &    |-  ( ( ph  /\  ( e  e.  B  /\  f  e.  B  /\  g  e.  B ) )  ->  ( g  e.  ( e J f )  <->  ( F `  g )  e.  (
 ( F `  e
 ) I ( F `
  f ) ) ) )   &    |-  ( ph  ->  H  e.  V )   &    |-  ( ph  ->  G  e. TarskiGE )   =>    |-  ( ph  ->  H  e. TarskiGE )
 
15.4  Geometry in Hilbert spaces
 
Syntaxcttg 23852 Function to convert an algebraic structure to a Tarski geometry.
 class toTG
 
Definitiondf-ttg 23853* Define a function converting a complex Hilbert space to a Tarski Geometry. It does so by equipping the structure with a betweenness operation. Note that because the scalar product is applied over the interval  ( 0 [,] 1 ), only spaces whose scalar field is a superset of that interval can be considered. (Contributed by Thierry Arnoux, 24-Mar-2019.)
 |- toTG  =  ( w  e.  _V  |->  [_ ( x  e.  ( Base `  w ) ,  y  e.  ( Base `  w )  |->  { z  e.  ( Base `  w )  |  E. k  e.  (
 0 [,] 1 ) ( z ( -g `  w ) x )  =  ( k ( .s `  w ) ( y ( -g `  w ) x ) ) }
 )  /  i ]_ ( ( w sSet  <. (Itv `  ndx ) ,  i >. ) sSet  <. (LineG `  ndx ) ,  ( x  e.  ( Base `  w ) ,  y  e.  ( Base `  w )  |->  { z  e.  ( Base `  w )  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) >. ) )
 
Theoremttgval 23854* Define a function to augment a complex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.)
 |-  G  =  (toTG `  H )   &    |-  B  =  (
 Base `  H )   &    |-  .-  =  ( -g `  H )   &    |-  .x. 
 =  ( .s `  H )   &    |-  I  =  (Itv `  G )   =>    |-  ( H  e.  V  ->  ( G  =  ( ( H sSet  <. (Itv `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  { z  e.  B  |  E. k  e.  (
 0 [,] 1 ) ( z  .-  x )  =  ( k  .x.  (
 y  .-  x )
 ) } ) >. ) sSet  <. (LineG `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  { z  e.  B  |  ( z  e.  ( x I y )  \/  x  e.  ( z I y )  \/  y  e.  ( x I z ) ) } ) >. )  /\  I  =  ( x  e.  B ,  y  e.  B  |->  { z  e.  B  |  E. k  e.  (
 0 [,] 1 ) ( z  .-  x )  =  ( k  .x.  (
 y  .-  x )
 ) } ) ) )
 
Theoremttglem 23855 Lemma for ttgbas 23856 and ttgvsca 23859. (Contributed by Thierry Arnoux, 15-Apr-2019.)
 |-  G  =  (toTG `  H )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  < ; 1 6   =>    |-  ( E `  H )  =  ( E `  G )
 
Theoremttgbas 23856 The base set of a complex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
 |-  G  =  (toTG `  H )   &    |-  B  =  (
 Base `  H )   =>    |-  B  =  (
 Base `  G )
 
Theoremttgplusg 23857 The addition operation of a complex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
 |-  G  =  (toTG `  H )   &    |-  .+  =  ( +g  `  H )   =>    |-  .+  =  ( +g  `  G )
 
Theoremttgsub 23858 The subtraction operation of a complex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
 |-  G  =  (toTG `  H )   &    |-  .-  =  ( -g `  H )   =>    |-  .-  =  ( -g `  G )
 
Theoremttgvsca 23859 The scalar product of a complex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
 |-  G  =  (toTG `  H )   &    |-  .x.  =  ( .s `  H )   =>    |-  .x.  =  ( .s `  G )
 
Theoremttgds 23860 The metric of a complex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
 |-  G  =  (toTG `  H )   &    |-  D  =  (
 dist `  H )   =>    |-  D  =  (
 dist `  G )
 
Theoremttgitvval 23861* Betweenness for a complex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
 |-  G  =  (toTG `  H )   &    |-  I  =  (Itv `  G )   &    |-  P  =  (
 Base `  H )   &    |-  .-  =  ( -g `  H )   &    |-  .x. 
 =  ( .s `  H )   =>    |-  ( ( H  e.  V  /\  X  e.  P  /\  Y  e.  P ) 
 ->  ( X I Y )  =  { z  e.  P  |  E. k  e.  ( 0 [,] 1
 ) ( z  .-  X )  =  (
 k  .x.  ( Y  .-  X ) ) }
 )
 
Theoremttgelitv 23862* Betweenness for a complex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
 |-  G  =  (toTG `  H )   &    |-  I  =  (Itv `  G )   &    |-  P  =  (
 Base `  H )   &    |-  .-  =  ( -g `  H )   &    |-  .x. 
 =  ( .s `  H )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  H  e.  V )   &    |-  ( ph  ->  Z  e.  P )   =>    |-  ( ph  ->  ( Z  e.  ( X I Y )  <->  E. k  e.  (
 0 [,] 1 ) ( Z  .-  X )  =  ( k  .x.  ( Y  .-  X ) ) ) )
 
Theoremttgbtwnid 23863 Any complex module equipped with the betweenness operation fulfills the identity of betweenness (Axiom A6). (Contributed by Thierry Arnoux, 26-Mar-2019.)
 |-  G  =  (toTG `  H )   &    |-  I  =  (Itv `  G )   &    |-  P  =  (
 Base `  H )   &    |-  .-  =  ( -g `  H )   &    |-  .x. 
 =  ( .s `  H )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  R  =  ( Base `  (Scalar `  H )
 )   &    |-  ( ph  ->  (
 0 [,] 1 )  C_  R )   &    |-  ( ph  ->  H  e. CMod )   &    |-  ( ph  ->  Y  e.  ( X I X ) )   =>    |-  ( ph  ->  X  =  Y )
 
Theoremttgcontlem1 23864 Lemma for % ttgcont (Contributed by Thierry Arnoux, 24-May-2019.)
 |-  G  =  (toTG `  H )   &    |-  I  =  (Itv `  G )   &    |-  P  =  (
 Base `  H )   &    |-  .-  =  ( -g `  H )   &    |-  .x. 
 =  ( .s `  H )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  R  =  ( Base `  (Scalar `  H )
 )   &    |-  ( ph  ->  (
 0 [,] 1 )  C_  R )   &    |-  .+  =  ( +g  `  H )   &    |-  ( ph  ->  H  e. CVec )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  N  e.  P )   &    |-  ( ph  ->  M  =/=  0 )   &    |-  ( ph  ->  K  =/=  0
 )   &    |-  ( ph  ->  K  =/=  1 )   &    |-  ( ph  ->  L  =/=  M )   &    |-  ( ph  ->  L  <_  ( M  /  K ) )   &    |-  ( ph  ->  L  e.  ( 0 [,] 1
 ) )   &    |-  ( ph  ->  K  e.  ( 0 [,] 1 ) )   &    |-  ( ph  ->  M  e.  (
 0 [,] L ) )   &    |-  ( ph  ->  ( X  .-  A )  =  ( K  .x.  ( Y  .-  A ) ) )   &    |-  ( ph  ->  ( X  .-  A )  =  ( M  .x.  ( N  .-  A ) ) )   &    |-  ( ph  ->  B  =  ( A  .+  ( L 
 .x.  ( N  .-  A ) ) ) )   =>    |-  ( ph  ->  B  e.  ( X I Y ) )
 
Theoremxmstrkgc 23865 Any metric space fulfills Tarski's geometry axioms of congruence. (Contributed by Thierry Arnoux, 13-Mar-2019.)
 |-  ( G  e.  *MetSp  ->  G  e. TarskiGC )
 
15.4.1  Geometry in the complex plane
 
Theoremcchhllem 23866* Lemma for chlbas and chlvsca . (Contributed by Thierry Arnoux, 15-Apr-2019.)
 |-  C  =  ( ( (subringAlg  ` fld ) `  RR ) sSet  <.
 ( .i `  ndx ) ,  ( x  e.  CC ,  y  e. 
 CC  |->  ( x  x.  ( * `  y ) ) ) >. )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  ( N  <  5  \/  8  <  N )   =>    |-  ( E ` fld )  =  ( E `  C )
 
15.4.2  Geometry in Euclidean spaces
 
15.4.2.1  Definition of the Euclidean space
 
Syntaxcee 23867 Declare the syntax for the Euclidean space generator.
 class  EE
 
Syntaxcbtwn 23868 Declare the syntax for the Euclidean betweenness predicate.
 class  Btwn
 
Syntaxccgr 23869 Declare the syntax for the Euclidean congruence predicate.
 class Cgr
 
Definitiondf-ee 23870 Define the Euclidean space generator. For details, see elee 23873. (Contributed by Scott Fenton, 3-Jun-2013.)
 |- 
 EE  =  ( n  e.  NN  |->  ( RR 
 ^m  ( 1 ... n ) ) )
 
Definitiondf-btwn 23871* Define the Euclidean betweenness predicate. For details, see brbtwn 23878. (Contributed by Scott Fenton, 3-Jun-2013.)
 |- 
 Btwn  =  `' { <. <. x ,  z >. ,  y >.  |  E. n  e.  NN  ( ( x  e.  ( EE `  n )  /\  z  e.  ( EE `  n )  /\  y  e.  ( EE `  n ) ) 
 /\  E. t  e.  (
 0 [,] 1 ) A. i  e.  ( 1 ... n ) ( y `
  i )  =  ( ( ( 1  -  t )  x.  ( x `  i
 ) )  +  (
 t  x.  ( z `
  i ) ) ) ) }
 
Definitiondf-cgr 23872* Define the Euclidean congruence predicate. For details, see brcgr 23879. (Contributed by Scott Fenton, 3-Jun-2013.)
 |- Cgr 
 =  { <. x ,  y >.  |  E. n  e.  NN  ( ( x  e.  ( ( EE
 `  n )  X.  ( EE `  n ) )  /\  y  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )  /\  sum_ i  e.  ( 1 ... n ) ( ( ( ( 1st `  x ) `  i )  -  ( ( 2nd `  x ) `  i ) ) ^ 2 )  = 
 sum_ i  e.  (
 1 ... n ) ( ( ( ( 1st `  y ) `  i
 )  -  ( ( 2nd `  y ) `  i ) ) ^
 2 ) ) }
 
Theoremelee 23873 Membership in a Euclidean space. We define Euclidean space here using Cartesian coordinates over 
N space. We later abstract away from this using Tarski's geometry axioms, so this exact definition is unimportant. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  ( N  e.  NN  ->  ( A  e.  ( EE `  N )  <->  A : ( 1
 ... N ) --> RR )
 )
 
Theoremmptelee 23874* A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.)
 |-  ( N  e.  NN  ->  ( ( k  e.  ( 1 ... N )  |->  ( A F B ) )  e.  ( EE `  N ) 
 <-> 
 A. k  e.  (
 1 ... N ) ( A F B )  e.  RR ) )
 
Theoremeleenn 23875 If  A is in  ( EE
`  N ), then  N is a natural. (Contributed by Scott Fenton, 1-Jul-2013.)
 |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )
 
Theoremeleei 23876 The forward direction of elee 23873. (Contributed by Scott Fenton, 1-Jul-2013.)
 |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ...
 N ) --> RR )
 
Theoremeedimeq 23877 A point belongs to at most one Euclidean space. (Contributed by Scott Fenton, 1-Jul-2013.)
 |-  ( ( A  e.  ( EE `  N ) 
 /\  A  e.  ( EE `  M ) ) 
 ->  N  =  M )
 
Theorembrbtwn 23878* The binary relationship form of the betweenness predicate. The statement  A  Btwn  <. B ,  C >. should be informally read as " A lies on a line segment between  B and  C. This exact definition is abstracted away by Tarski's geometry axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( A  Btwn  <. B ,  C >. 
 <-> 
 E. t  e.  (
 0 [,] 1 ) A. i  e.  ( 1 ... N ) ( A `
  i )  =  ( ( ( 1  -  t )  x.  ( B `  i
 ) )  +  (
 t  x.  ( C `
  i ) ) ) ) )
 
Theorembrcgr 23879* The binary relationship form of the congruence predicate. The statement  <. A ,  B >.Cgr <. C ,  D >. should be read informally as "the  N dimensional point  A is as far from  B as  C is from  D, or "the line segment  A B is congruent to the line segment  C D. This particular definition is encapsulated by Tarski's axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.Cgr <. C ,  D >.  <->  sum_ i  e.  ( 1 ...
 N ) ( ( ( A `  i
 )  -  ( B `
  i ) ) ^ 2 )  = 
 sum_ i  e.  (
 1 ... N ) ( ( ( C `  i )  -  ( D `  i ) ) ^ 2 ) ) )
 
Theoremfveere 23880 The function value of a point is a real. (Contributed by Scott Fenton, 10-Jun-2013.)
 |-  ( ( A  e.  ( EE `  N ) 
 /\  I  e.  (
 1 ... N ) ) 
 ->  ( A `  I
 )  e.  RR )
 
Theoremfveecn 23881 The function value of a point is a complex. (Contributed by Scott Fenton, 10-Jun-2013.)
 |-  ( ( A  e.  ( EE `  N ) 
 /\  I  e.  (
 1 ... N ) ) 
 ->  ( A `  I
 )  e.  CC )
 
Theoremeqeefv 23882* Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.)
 |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 ->  ( A  =  B  <->  A. i  e.  ( 1
 ... N ) ( A `  i )  =  ( B `  i ) ) )
 
Theoremeqeelen 23883* Two points are equal iff the square of the distance between them is zero. (Contributed by Scott Fenton, 10-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 ->  ( A  =  B  <->  sum_
 i  e.  ( 1
 ... N ) ( ( ( A `  i )  -  ( B `  i ) ) ^ 2 )  =  0 ) )
 
Theorembrbtwn2 23884* Alternate characterization of betweenness, with no existential quantifiers. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( A  Btwn  <. B ,  C >. 
 <->  ( A. i  e.  ( 1 ... N ) ( ( ( B `  i )  -  ( A `  i ) )  x.  ( ( C `  i )  -  ( A `  i ) ) )  <_  0  /\  A. i  e.  ( 1
 ... N ) A. j  e.  ( 1 ... N ) ( ( ( B `  i
 )  -  ( A `
  i ) )  x.  ( ( C `
  j )  -  ( A `  j ) ) )  =  ( ( ( B `  j )  -  ( A `  j ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) ) ) ) )
 
Theoremcolinearalglem1 23885 Lemma for colinearalg 23889. Expand out a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC ) )  ->  (
 ( ( B  -  A )  x.  ( F  -  D ) )  =  ( ( E  -  D )  x.  ( C  -  A ) )  <->  ( ( B  x.  F )  -  ( ( A  x.  F )  +  ( B  x.  D ) ) )  =  ( ( C  x.  E )  -  ( ( A  x.  E )  +  ( C  x.  D ) ) ) ) )
 
Theoremcolinearalglem2 23886* Lemma for colinearalg 23889. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( B `  i
 )  -  ( A `
  i ) )  x.  ( ( C `
  j )  -  ( A `  j ) ) )  =  ( ( ( B `  j )  -  ( A `  j ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) )  <->  A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( C `  i
 )  -  ( B `
  i ) )  x.  ( ( A `
  j )  -  ( B `  j ) ) )  =  ( ( ( C `  j )  -  ( B `  j ) )  x.  ( ( A `
  i )  -  ( B `  i ) ) ) ) )
 
Theoremcolinearalglem3 23887* Lemma for colinearalg 23889. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( B `  i
 )  -  ( A `
  i ) )  x.  ( ( C `
  j )  -  ( A `  j ) ) )  =  ( ( ( B `  j )  -  ( A `  j ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) )  <->  A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( A `  i
 )  -  ( C `
  i ) )  x.  ( ( B `
  j )  -  ( C `  j ) ) )  =  ( ( ( A `  j )  -  ( C `  j ) )  x.  ( ( B `
  i )  -  ( C `  i ) ) ) ) )
 
Theoremcolinearalglem4 23888* Lemma for colinearalg 23889. Prove a disjunction that will be needed in the final proof. (Contributed by Scott Fenton, 27-Jun-2013.)
 |-  ( ( ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  K  e.  RR )  ->  ( A. i  e.  ( 1 ... N ) ( ( ( ( K  x.  ( ( C `  i )  -  ( A `  i ) ) )  +  ( A `
  i ) )  -  ( A `  i ) )  x.  ( ( C `  i )  -  ( A `  i ) ) )  <_  0  \/  A. i  e.  ( 1
 ... N ) ( ( ( C `  i )  -  (
 ( K  x.  (
 ( C `  i
 )  -  ( A `
  i ) ) )  +  ( A `
  i ) ) )  x.  ( ( A `  i )  -  ( ( K  x.  ( ( C `
  i )  -  ( A `  i ) ) )  +  ( A `  i ) ) ) )  <_  0  \/  A. i  e.  (
 1 ... N ) ( ( ( A `  i )  -  ( C `  i ) )  x.  ( ( ( K  x.  ( ( C `  i )  -  ( A `  i ) ) )  +  ( A `  i ) )  -  ( C `  i ) ) )  <_  0
 ) )
 
Theoremcolinearalg 23889* An algebraic characterization of colinearity. Note the similarity to brbtwn2 23884. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. )  <->  A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( B `  i
 )  -  ( A `
  i ) )  x.  ( ( C `
  j )  -  ( A `  j ) ) )  =  ( ( ( B `  j )  -  ( A `  j ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) ) ) )
 
Theoremeleesub 23890* Membership of a subtraction mapping in a Euclidean space. (Contributed by Scott Fenton, 17-Jul-2013.)
 |-  C  =  ( i  e.  ( 1 ...
 N )  |->  ( ( A `  i )  -  ( B `  i ) ) )   =>    |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 ->  C  e.  ( EE
 `  N ) )
 
Theoremeleesubd 23891* Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 23890. (Contributed by Scott Fenton, 17-Jul-2013.)
 |-  ( ph  ->  C  =  ( i  e.  (
 1 ... N )  |->  ( ( A `  i
 )  -  ( B `
  i ) ) ) )   =>    |-  ( ( ph  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 ->  C  e.  ( EE
 `  N ) )
 
15.4.2.2  Tarski's axioms for geometry for the Euclidean space
 
Theoremaxdimuniq 23892 The unique dimension axiom. If a point is in  N dimensional space and in  M dimensional space, then  N  =  M. This axiom is not traditionally presented with Tarski's axioms, but we require it here as we are considering spaces in arbitrary dimensions. (Contributed by Scott Fenton, 24-Sep-2013.)
 |-  ( ( ( N  e.  NN  /\  A  e.  ( EE `  N ) )  /\  ( M  e.  NN  /\  A  e.  ( EE `  M ) ) )  ->  N  =  M )
 
Theoremaxcgrrflx 23893  A is as far from  B as  B is from  A. Axiom A1 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.Cgr <. B ,  A >. )
 
Theoremaxcgrtr 23894 Congruence is transitive. Axiom A2 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-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 ) ) )  ->  ( ( <. A ,  B >.Cgr <. C ,  D >.  /\  <. A ,  B >.Cgr
 <. E ,  F >. ) 
 ->  <. C ,  D >.Cgr
 <. E ,  F >. ) )
 
Theoremaxcgrid 23895 If there is no distance between  A and  B, then  A  =  B. Axiom A3 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  C >.  ->  A  =  B )
 )
 
Theoremaxsegconlem1 23896* Lemma for axsegcon 23906. Handle the degenerate case. (Contributed by Scott Fenton, 7-Jun-2013.)
 |-  ( ( A  =  B  /\  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) )  ->  E. x  e.  ( EE `  N ) E. t  e.  ( 0 [,] 1 ) ( A. i  e.  ( 1 ... N ) ( B `
  i )  =  ( ( ( 1  -  t )  x.  ( A `  i
 ) )  +  (
 t  x.  ( x `
  i ) ) )  /\  sum_ i  e.  ( 1 ... N ) ( ( ( B `  i )  -  ( x `  i ) ) ^
 2 )  =  sum_ i  e.  ( 1 ...
 N ) ( ( ( C `  i
 )  -  ( D `
  i ) ) ^ 2 ) ) )
 
Theoremaxsegconlem2 23897* Lemma for axsegcon 23906. Show that the square of the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  ( 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^
 2 )   =>    |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 ->  S  e.  RR )
 
Theoremaxsegconlem3 23898* Lemma for axsegcon 23906. Show that the square of the distance between two points is nonnegative. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  ( 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^
 2 )   =>    |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 ->  0  <_  S )
 
Theoremaxsegconlem4 23899* Lemma for axsegcon 23906. Show that the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  ( 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^
 2 )   =>    |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 ->  ( sqr `  S )  e.  RR )
 
Theoremaxsegconlem5 23900* Lemma for axsegcon 23906. Show that the distance between two points is nonnegative. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  ( 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^
 2 )   =>    |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 ->  0  <_  ( sqr `  S ) )
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