Definition df-trkg 25352

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 2656.

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 (TarskiG_C)
• A3 Identity for the congruence relation (TarskiG_C)
• A4 Axiom of segment construction (TarskiG_CB)
• A5 5-segment axiom (TarskiG_CB)
• A6 Identity for the betweenness relation (TarskiG_B)
• A7 Axiom of Pasch (TarskiG_B)
• A8 Lower dimension axiom (Dim_TarskiG≥ ` 2 )
• A9 Upper dimension axiom ( _V \ (Dim<sub>TarskiG</sub>≥ ` 3 ) )
• A10 Euclid's axiom (TarskiG_E)
• A11 Axiom of continuity (TarskiG_B)

Our definition is split into 5 parts:

• congruence axioms TarskiG_C (which metric spaces fulfill)
• betweenness axioms TarskiG_B
• congruence and betweenness axioms TarskiG_CB
• upper and lower dimension axioms Dim_TarskiG≥
• axiom of Euclid / parallel postulate TarskiG_E

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

Step	Hyp	Ref	Expression
1		cstrkg 25329	. 2 class TarskiG
2		cstrkgc 25330	. . . 4 class TarskiGC
3		cstrkgb 25331	. . . 4 class TarskiGB
4	2, 3	cin 3573	. . 3 class (TarskiGC i^i TarskiGB )
5		cstrkgcb 25332	. . . 4 class TarskiGCB
6		vf	. . . . . . . . . 10 setvar f
7	6	cv 1482	. . . . . . . . 9 class f
8		clng 25336	. . . . . . . . 9 class LineG
9	7, 8	cfv 5888	. . . . . . . 8 class (LineG ` f )
10		vx	. . . . . . . . 9 setvar x
11		vy	. . . . . . . . 9 setvar y
12		vp	. . . . . . . . . 10 setvar p
13	12	cv 1482	. . . . . . . . 9 class p
14	10	cv 1482	. . . . . . . . . . 11 class x
15	14	csn 4177	. . . . . . . . . 10 class { x }
16	13, 15	cdif 3571	. . . . . . . . 9 class ( p \ { x } )
17		vz	. . . . . . . . . . . . 13 setvar z
18	17	cv 1482	. . . . . . . . . . . 12 class z
19	11	cv 1482	. . . . . . . . . . . . 13 class y
20		vi	. . . . . . . . . . . . . 14 setvar i
21	20	cv 1482	. . . . . . . . . . . . 13 class i
22	14, 19, 21	co 6650	. . . . . . . . . . . 12 class ( x i y )
23	18, 22	wcel 1990	. . . . . . . . . . 11 wff z e. ( x i y )
24	18, 19, 21	co 6650	. . . . . . . . . . . 12 class ( z i y )
25	14, 24	wcel 1990	. . . . . . . . . . 11 wff x e. ( z i y )
26	14, 18, 21	co 6650	. . . . . . . . . . . 12 class ( x i z )
27	19, 26	wcel 1990	. . . . . . . . . . 11 wff y e. ( x i z )
28	23, 25, 27	w3o 1036	. . . . . . . . . 10 wff ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) )
29	28, 17, 13	crab 2916	. . . . . . . . 9 class { z e. p | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) }
30	10, 11, 13, 16, 29	cmpt2 6652	. . . . . . . 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 ) ) } )
31	9, 30	wceq 1483	. . . . . . 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 25335	. . . . . . . 8 class Itv
33	7, 32	cfv 5888	. . . . . . 7 class (Itv ` f )
34	31, 20, 33	wsbc 3435	. . . . . 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 15857	. . . . . . 7 class Base
36	7, 35	cfv 5888	. . . . . 6 class ( Base ` f )
37	34, 12, 36	wsbc 3435	. . . . 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 ) ) } )
38	37, 6	cab 2608	. . . 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 ) ) } ) }
39	5, 38	cin 3573	. . 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 ) ) } ) } )
40	4, 39	cin 3573	. 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 ) ) } ) } ) )
41	1, 40	wceq 1483	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 ) ) } ) } ) )

This definition is referenced by:  axtgcgrrflx  25361  axtgcgrid  25362  axtgsegcon  25363  axtg5seg  25364  axtgbtwnid  25365  axtgpasch  25366  axtgcont1  25367  tglng  25441  f1otrg  25751  eengtrkg  25865