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Definition df-trkg 23722
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 2484.

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 23697 . 2  class TarskiG
2 cstrkgc 23698 . . . 4  class TarskiGC
3 cstrkgb 23699 . . . 4  class TarskiGB
42, 3cin 3460 . . 3  class  (TarskiGC  i^i TarskiGB )
5 cstrkgcb 23700 . . . 4  class TarskiGCB
6 vf . . . . . . . . . 10  setvar  f
76cv 1382 . . . . . . . . 9  class  f
8 clng 23705 . . . . . . . . 9  class LineG
97, 8cfv 5578 . . . . . . . 8  class  (LineG `  f )
10 vx . . . . . . . . 9  setvar  x
11 vy . . . . . . . . 9  setvar  y
12 vp . . . . . . . . . 10  setvar  p
1312cv 1382 . . . . . . . . 9  class  p
1410cv 1382 . . . . . . . . . . 11  class  x
1514csn 4014 . . . . . . . . . 10  class  { x }
1613, 15cdif 3458 . . . . . . . . 9  class  ( p 
\  { x }
)
17 vz . . . . . . . . . . . . 13  setvar  z
1817cv 1382 . . . . . . . . . . . 12  class  z
1911cv 1382 . . . . . . . . . . . . 13  class  y
20 vi . . . . . . . . . . . . . 14  setvar  i
2120cv 1382 . . . . . . . . . . . . 13  class  i
2214, 19, 21co 6281 . . . . . . . . . . . 12  class  ( x i y )
2318, 22wcel 1804 . . . . . . . . . . 11  wff  z  e.  ( x i y )
2418, 19, 21co 6281 . . . . . . . . . . . 12  class  ( z i y )
2514, 24wcel 1804 . . . . . . . . . . 11  wff  x  e.  ( z i y )
2614, 18, 21co 6281 . . . . . . . . . . . 12  class  ( x i z )
2719, 26wcel 1804 . . . . . . . . . . 11  wff  y  e.  ( x i z )
2823, 25, 27w3o 973 . . . . . . . . . 10  wff  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) )
2928, 17, 13crab 2797 . . . . . . . . 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 6283 . . . . . . . 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 1383 . . . . . . 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 23704 . . . . . . . 8  class Itv
337, 32cfv 5578 . . . . . . 7  class  (Itv `  f )
3431, 20, 33wsbc 3313 . . . . . 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 14509 . . . . . . 7  class  Base
367, 35cfv 5578 . . . . . 6  class  ( Base `  f )
3734, 12, 36wsbc 3313 . . . . 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 2428 . . . 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 3460 . . 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 3460 . 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 1383 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  23731  axtgcgrid  23732  axtgsegcon  23733  axtg5seg  23734  axtgbtwnid  23735  axtgpasch  23736  axtgcont1  23737  tglng  23805  f1otrg  24046  eengtrkg  24160
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