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Definition df-trkg 24549
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 2495.

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 24526 . 2  class TarskiG
2 cstrkgc 24527 . . . 4  class TarskiGC
3 cstrkgb 24528 . . . 4  class TarskiGB
42, 3cin 3414 . . 3  class  (TarskiGC  i^i TarskiGB )
5 cstrkgcb 24529 . . . 4  class TarskiGCB
6 vf . . . . . . . . . 10  setvar  f
76cv 1453 . . . . . . . . 9  class  f
8 clng 24533 . . . . . . . . 9  class LineG
97, 8cfv 5600 . . . . . . . 8  class  (LineG `  f )
10 vx . . . . . . . . 9  setvar  x
11 vy . . . . . . . . 9  setvar  y
12 vp . . . . . . . . . 10  setvar  p
1312cv 1453 . . . . . . . . 9  class  p
1410cv 1453 . . . . . . . . . . 11  class  x
1514csn 3979 . . . . . . . . . 10  class  { x }
1613, 15cdif 3412 . . . . . . . . 9  class  ( p 
\  { x }
)
17 vz . . . . . . . . . . . . 13  setvar  z
1817cv 1453 . . . . . . . . . . . 12  class  z
1911cv 1453 . . . . . . . . . . . . 13  class  y
20 vi . . . . . . . . . . . . . 14  setvar  i
2120cv 1453 . . . . . . . . . . . . 13  class  i
2214, 19, 21co 6314 . . . . . . . . . . . 12  class  ( x i y )
2318, 22wcel 1897 . . . . . . . . . . 11  wff  z  e.  ( x i y )
2418, 19, 21co 6314 . . . . . . . . . . . 12  class  ( z i y )
2514, 24wcel 1897 . . . . . . . . . . 11  wff  x  e.  ( z i y )
2614, 18, 21co 6314 . . . . . . . . . . . 12  class  ( x i z )
2719, 26wcel 1897 . . . . . . . . . . 11  wff  y  e.  ( x i z )
2823, 25, 27w3o 990 . . . . . . . . . 10  wff  ( z  e.  ( x i y )  \/  x  e.  ( z i y )  \/  y  e.  ( x i z ) )
2928, 17, 13crab 2752 . . . . . . . . 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 6316 . . . . . . . 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 1454 . . . . . . 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 24532 . . . . . . . 8  class Itv
337, 32cfv 5600 . . . . . . 7  class  (Itv `  f )
3431, 20, 33wsbc 3278 . . . . . 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 15169 . . . . . . 7  class  Base
367, 35cfv 5600 . . . . . 6  class  ( Base `  f )
3734, 12, 36wsbc 3278 . . . . 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 2447 . . . 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 3414 . . 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 3414 . 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 1454 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  24558  axtgcgrid  24559  axtgsegcon  24560  axtg5seg  24561  axtgbtwnid  24562  axtgpasch  24563  axtgcont1  24564  tglng  24639  f1otrg  24949  eengtrkg  25063
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