Definition df-trkgc 24544

Description: Define the class of geometries fulfilling the congruence axioms of reflexivity, identity and transitivity. These are axioms A1 to A3 of [Schwabhauser] p. 10. With our distance based notation for congruence, transitivity of congruence boils down to transitivity of equality and is already given by eqtr 2480, so it is not listed in this definition. (Contributed by Thierry Arnoux, 24-Aug-2017.)

Assertion

Ref	Expression
df-trkgc	|- TarskiGC = { f | [. ( Base ` f ) / p ]. [. ( dist ` f ) / d ]. ( A. x e. p A. y e. p ( x d y ) = ( y d x ) /\ A. x e. p A. y e. p A. z e. p ( ( x d y ) = ( z d z ) -> x = y ) ) }

Distinct variable group:   f, d, p, x, y, z

Detailed syntax breakdown of Definition df-trkgc

Step	Hyp	Ref	Expression
1		cstrkgc 24527	. 2 class TarskiGC
2		vx	. . . . . . . . . . 11 setvar x
3	2	cv 1453	. . . . . . . . . 10 class x
4		vy	. . . . . . . . . . 11 setvar y
5	4	cv 1453	. . . . . . . . . 10 class y
6		vd	. . . . . . . . . . 11 setvar d
7	6	cv 1453	. . . . . . . . . 10 class d
8	3, 5, 7	co 6314	. . . . . . . . 9 class ( x d y )
9	5, 3, 7	co 6314	. . . . . . . . 9 class ( y d x )
10	8, 9	wceq 1454	. . . . . . . 8 wff ( x d y ) = ( y d x )
11		vp	. . . . . . . . 9 setvar p
12	11	cv 1453	. . . . . . . 8 class p
13	10, 4, 12	wral 2748	. . . . . . 7 wff A. y e. p ( x d y ) = ( y d x )
14	13, 2, 12	wral 2748	. . . . . 6 wff A. x e. p A. y e. p ( x d y ) = ( y d x )
15		vz	. . . . . . . . . . . . 13 setvar z
16	15	cv 1453	. . . . . . . . . . . 12 class z
17	16, 16, 7	co 6314	. . . . . . . . . . 11 class ( z d z )
18	8, 17	wceq 1454	. . . . . . . . . 10 wff ( x d y ) = ( z d z )
19	2, 4	weq 1801	. . . . . . . . . 10 wff x = y
20	18, 19	wi 4	. . . . . . . . 9 wff ( ( x d y ) = ( z d z ) -> x = y )
21	20, 15, 12	wral 2748	. . . . . . . 8 wff A. z e. p ( ( x d y ) = ( z d z ) -> x = y )
22	21, 4, 12	wral 2748	. . . . . . 7 wff A. y e. p A. z e. p ( ( x d y ) = ( z d z ) -> x = y )
23	22, 2, 12	wral 2748	. . . . . 6 wff A. x e. p A. y e. p A. z e. p ( ( x d y ) = ( z d z ) -> x = y )
24	14, 23	wa 375	. . . . 5 wff ( A. x e. p A. y e. p ( x d y ) = ( y d x ) /\ A. x e. p A. y e. p A. z e. p ( ( x d y ) = ( z d z ) -> x = y ) )
25		vf	. . . . . . 7 setvar f
26	25	cv 1453	. . . . . 6 class f
27		cds 15247	. . . . . 6 class dist
28	26, 27	cfv 5600	. . . . 5 class ( dist ` f )
29	24, 6, 28	wsbc 3278	. . . 4 wff [. ( dist ` f ) / d ]. ( A. x e. p A. y e. p ( x d y ) = ( y d x ) /\ A. x e. p A. y e. p A. z e. p ( ( x d y ) = ( z d z ) -> x = y ) )
30		cbs 15169	. . . . 5 class Base
31	26, 30	cfv 5600	. . . 4 class ( Base ` f )
32	29, 11, 31	wsbc 3278	. . 3 wff [. ( Base ` f ) / p ]. [. ( dist ` f ) / d ]. ( A. x e. p A. y e. p ( x d y ) = ( y d x ) /\ A. x e. p A. y e. p A. z e. p ( ( x d y ) = ( z d z ) -> x = y ) )
33	32, 25	cab 2447	. 2 class { f | [. ( Base ` f ) / p ]. [. ( dist ` f ) / d ]. ( A. x e. p A. y e. p ( x d y ) = ( y d x ) /\ A. x e. p A. y e. p A. z e. p ( ( x d y ) = ( z d z ) -> x = y ) ) }
34	1, 33	wceq 1454	1 wff TarskiGC = { f | [. ( Base ` f ) / p ]. [. ( dist ` f ) / d ]. ( A. x e. p A. y e. p ( x d y ) = ( y d x ) /\ A. x e. p A. y e. p A. z e. p ( ( x d y ) = ( z d z ) -> x = y ) ) }

This definition is referenced by:  istrkgc  24550