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Definition df-trkg 22914
Description: Define the class of Tarski geometries. A Tarski geometry is a set of points, equipped with a betweenness relation (denoting that a point lies on a line segment between two other points) and a congruence relation (denoting equality of line segment lengths). Here, we are using the following:
  • for congruence,  ( x  .-  y )  =  ( z  .-  w ) where  .-  =  ( dist `  W )
  • for betweenness,  y  e.  ( x I z ), where  I  =  (Itv `  W )
With this definition, the axiom A2 is actually equivalent to the transitivity of addition, eqtrd 2473.

Tarski originally had more axioms, but later reduced his list to 11:

  • A1 A kind of reflexivity for the congruence relation (TarskiGC)
  • A2 Transitivity for the congruence relation (TarskiGC)
  • A3 Identity for the congruence relation (TarskiGC)
  • A4 Axiom of segment construction (TarskiGBC)
  • A5 5-segment axiom (TarskiGBC)
  • A6 Identity for the betweenness relation (TarskiGB)
  • A7 Axiom of Pasch (TarskiGB)
  • A8 Lower dimension axiom (TarskiG2D)
  • A9 Upper dimension axiom (TarskiG2D)
  • A10 Euclid's axiom (TarskiGE)
  • A11 Axiom of continuity (TarskiGB)
Our definition is split into 5 parts:
  • congruence axioms TarskiGC (which metric spaces fulfill)
  • betweenness axioms TarskiGB
  • congruence and betweenness axioms TarskiGCB
  • upper and lower dimension axioms TarskiG2D
  • axiom of Euclid / parallel postulate TarskiGE

So our definition of a Tarskian Geometry includes the 3 axioms for the quaternary congruence relation (A1, A2, A3), the 3 axioms for the ternary betweenness relation (A6, A7, A11), and the 2 axioms of compatibility of the congruence and the betweenness relations (A4,A5).

It does not include Euclid's axiom A10, nor the 2-dimensional axioms A8 (Lower dimension axiom) and A9 (Upper dimension axiom) so the number of dimensions of the geometry it formalizes is not constrained.

Considering A2 as one of the 3 axioms for the quaternary congruence relation is somewhat conventional, because the transitivity of the congruence relation is automatically given by our choice to take the distance as this congruence relation in our definition of Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) (Revised by Thierry Arnoux, 27-Apr-2019.)

Assertion
Ref Expression
df-trkg  |- TarskiG  =  ( (TarskiGC  i^i TarskiGB )  i^i  (TarskiGCB  i^i  {
f  |  [. ( Base `  f )  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p  \  {
x } )  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) } ) )
Distinct variable group:    f, p, i, x, y, z

Detailed syntax breakdown of Definition df-trkg
StepHypRef Expression
1 cstrkg 22887 . 2  class TarskiG
2 cstrkgc 22888 . . . 4  class TarskiGC
3 cstrkgb 22889 . . . 4  class TarskiGB
42, 3cin 3325 . . 3  class  (TarskiGC  i^i TarskiGB )
5 cstrkgcb 22890 . . . 4  class TarskiGCB
6 vf . . . . . . . . . 10  setvar  f
76cv 1368 . . . . . . . . 9  class  f
8 clng 22896 . . . . . . . . 9  class LineG
97, 8cfv 5416 . . . . . . . 8  class  (LineG `  f )
10 vx . . . . . . . . 9  setvar  x
11 vy . . . . . . . . 9  setvar  y
12 vp . . . . . . . . . 10  setvar  p
1312cv 1368 . . . . . . . . 9  class  p
1410cv 1368 . . . . . . . . . . 11  class  x
1514csn 3875 . . . . . . . . . 10  class  { x }
1613, 15cdif 3323 . . . . . . . . 9  class  ( p 
\  { x }
)
17 vz . . . . . . . . . . . . 13  setvar  z
1817cv 1368 . . . . . . . . . . . 12  class  z
1911cv 1368 . . . . . . . . . . . . 13  class  y
20 vi . . . . . . . . . . . . . 14  setvar  i
2120cv 1368 . . . . . . . . . . . . 13  class  i
2214, 19, 21co 6089 . . . . . . . . . . . 12  class  ( x i y )
2318, 22wcel 1756 . . . . . . . . . . 11  wff  z  e.  ( x i y )
2418, 19, 21co 6089 . . . . . . . . . . . 12  class  ( z i y )
2514, 24wcel 1756 . . . . . . . . . . 11  wff  x  e.  ( z i y )
2614, 18, 21co 6089 . . . . . . . . . . . 12  class  ( x i z )
2719, 26wcel 1756 . . . . . . . . . . 11  wff  y  e.  ( x i z )
2823, 25, 27w3o 964 . . . . . . . . . 10  wff  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) )
2928, 17, 13crab 2717 . . . . . . . . 9  class  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) }
3010, 11, 13, 16, 29cmpt2 6091 . . . . . . . 8  class  ( x  e.  p ,  y  e.  ( p  \  { x } ) 
|->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } )
319, 30wceq 1369 . . . . . . 7  wff  (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p 
\  { x }
)  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } )
32 citv 22895 . . . . . . . 8  class Itv
337, 32cfv 5416 . . . . . . 7  class  (Itv `  f )
3431, 20, 33wsbc 3184 . . . . . 6  wff  [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p  \  {
x } )  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } )
35 cbs 14172 . . . . . . 7  class  Base
367, 35cfv 5416 . . . . . 6  class  ( Base `  f )
3734, 12, 36wsbc 3184 . . . . 5  wff  [. ( Base `  f )  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p  \  {
x } )  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } )
3837, 6cab 2427 . . . 4  class  { f  |  [. ( Base `  f )  /  p ]. [. (Itv `  f
)  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  (
p  \  { x } )  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) }
395, 38cin 3325 . . 3  class  (TarskiGCB  i^i  {
f  |  [. ( Base `  f )  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p  \  {
x } )  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) } )
404, 39cin 3325 . 2  class  ( (TarskiGC  i^i TarskiGB )  i^i  (TarskiGCB  i^i  { f  | 
[. ( Base `  f
)  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p 
\  { x }
)  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) } ) )
411, 40wceq 1369 1  wff TarskiG  =  ( (TarskiGC  i^i TarskiGB )  i^i  (TarskiGCB  i^i  {
f  |  [. ( Base `  f )  /  p ]. [. (Itv `  f )  /  i ]. (LineG `  f )  =  ( x  e.  p ,  y  e.  ( p  \  {
x } )  |->  { z  e.  p  |  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) ) } ) } ) )
Colors of variables: wff setvar class
This definition is referenced by:  axtgcgrrflx  22921  axtgcgrid  22922  axtgsegcon  22923  axtg5seg  22924  axtgbtwnid  22925  axtgpasch  22926  axtgcont1  22927  tglng  22978  f1otrg  23115  eengtrkg  23229
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