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Definition df-trkg 24443
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 2462.

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 (TarskiGCB)
  • A5 5-segment axiom (TarskiGCB)
  • A6 Identity for the betweenness relation (TarskiGB)
  • A7 Axiom of Pasch (TarskiGB)
  • A8 Lower dimension axiom  (DimTarskiG `  2 )
  • A9 Upper dimension axiom  ( _V  \  (DimTarskiG `  3 ) )
  • 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 DimTarskiG
  • 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 24420 . 2  class TarskiG
2 cstrkgc 24421 . . . 4  class TarskiGC
3 cstrkgb 24422 . . . 4  class TarskiGB
42, 3cin 3378 . . 3  class  (TarskiGC  i^i TarskiGB )
5 cstrkgcb 24423 . . . 4  class TarskiGCB
6 vf . . . . . . . . . 10  setvar  f
76cv 1436 . . . . . . . . 9  class  f
8 clng 24427 . . . . . . . . 9  class LineG
97, 8cfv 5544 . . . . . . . 8  class  (LineG `  f )
10 vx . . . . . . . . 9  setvar  x
11 vy . . . . . . . . 9  setvar  y
12 vp . . . . . . . . . 10  setvar  p
1312cv 1436 . . . . . . . . 9  class  p
1410cv 1436 . . . . . . . . . . 11  class  x
1514csn 3941 . . . . . . . . . 10  class  { x }
1613, 15cdif 3376 . . . . . . . . 9  class  ( p 
\  { x }
)
17 vz . . . . . . . . . . . . 13  setvar  z
1817cv 1436 . . . . . . . . . . . 12  class  z
1911cv 1436 . . . . . . . . . . . . 13  class  y
20 vi . . . . . . . . . . . . . 14  setvar  i
2120cv 1436 . . . . . . . . . . . . 13  class  i
2214, 19, 21co 6249 . . . . . . . . . . . 12  class  ( x i y )
2318, 22wcel 1872 . . . . . . . . . . 11  wff  z  e.  ( x i y )
2418, 19, 21co 6249 . . . . . . . . . . . 12  class  ( z i y )
2514, 24wcel 1872 . . . . . . . . . . 11  wff  x  e.  ( z i y )
2614, 18, 21co 6249 . . . . . . . . . . . 12  class  ( x i z )
2719, 26wcel 1872 . . . . . . . . . . 11  wff  y  e.  ( x i z )
2823, 25, 27w3o 981 . . . . . . . . . 10  wff  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) )
2928, 17, 13crab 2718 . . . . . . . . 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 6251 . . . . . . . 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 1437 . . . . . . 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 24426 . . . . . . . 8  class Itv
337, 32cfv 5544 . . . . . . 7  class  (Itv `  f )
3431, 20, 33wsbc 3242 . . . . . 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 15064 . . . . . . 7  class  Base
367, 35cfv 5544 . . . . . 6  class  ( Base `  f )
3734, 12, 36wsbc 3242 . . . . 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 2414 . . . 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 3378 . . 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 3378 . 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 1437 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  24452  axtgcgrid  24453  axtgsegcon  24454  axtg5seg  24455  axtgbtwnid  24456  axtgpasch  24457  axtgcont1  24458  tglng  24533  f1otrg  24843  eengtrkg  24957
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