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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
StepHypRef Expression
1 cstrkgc 24527 . 2  class TarskiGC
2 vx . . . . . . . . . . 11  setvar  x
32cv 1453 . . . . . . . . . 10  class  x
4 vy . . . . . . . . . . 11  setvar  y
54cv 1453 . . . . . . . . . 10  class  y
6 vd . . . . . . . . . . 11  setvar  d
76cv 1453 . . . . . . . . . 10  class  d
83, 5, 7co 6314 . . . . . . . . 9  class  ( x d y )
95, 3, 7co 6314 . . . . . . . . 9  class  ( y d x )
108, 9wceq 1454 . . . . . . . 8  wff  ( x d y )  =  ( y d x )
11 vp . . . . . . . . 9  setvar  p
1211cv 1453 . . . . . . . 8  class  p
1310, 4, 12wral 2748 . . . . . . 7  wff  A. y  e.  p  ( x
d y )  =  ( y d x )
1413, 2, 12wral 2748 . . . . . 6  wff  A. x  e.  p  A. y  e.  p  ( x
d y )  =  ( y d x )
15 vz . . . . . . . . . . . . 13  setvar  z
1615cv 1453 . . . . . . . . . . . 12  class  z
1716, 16, 7co 6314 . . . . . . . . . . 11  class  ( z d z )
188, 17wceq 1454 . . . . . . . . . 10  wff  ( x d y )  =  ( z d z )
192, 4weq 1801 . . . . . . . . . 10  wff  x  =  y
2018, 19wi 4 . . . . . . . . 9  wff  ( ( x d y )  =  ( z d z )  ->  x  =  y )
2120, 15, 12wral 2748 . . . . . . . 8  wff  A. z  e.  p  ( (
x d y )  =  ( z d z )  ->  x  =  y )
2221, 4, 12wral 2748 . . . . . . 7  wff  A. y  e.  p  A. z  e.  p  ( (
x d y )  =  ( z d z )  ->  x  =  y )
2322, 2, 12wral 2748 . . . . . 6  wff  A. x  e.  p  A. y  e.  p  A. z  e.  p  ( (
x d y )  =  ( z d z )  ->  x  =  y )
2414, 23wa 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
2625cv 1453 . . . . . 6  class  f
27 cds 15247 . . . . . 6  class  dist
2826, 27cfv 5600 . . . . 5  class  ( dist `  f )
2924, 6, 28wsbc 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
3126, 30cfv 5600 . . . 4  class  ( Base `  f )
3229, 11, 31wsbc 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 ) )
3332, 25cab 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 )
) }
341, 33wceq 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 )
) }
Colors of variables: wff setvar class
This definition is referenced by:  istrkgc  24550
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