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Definition df-trkgc 25544
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 2777, 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](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
Distinct variable group:   𝑓,𝑑,𝑝,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-trkgc
StepHypRef Expression
1 cstrkgc 25527 . 2 class TarskiGC
2 vx . . . . . . . . . . 11 setvar 𝑥
32cv 1629 . . . . . . . . . 10 class 𝑥
4 vy . . . . . . . . . . 11 setvar 𝑦
54cv 1629 . . . . . . . . . 10 class 𝑦
6 vd . . . . . . . . . . 11 setvar 𝑑
76cv 1629 . . . . . . . . . 10 class 𝑑
83, 5, 7co 6811 . . . . . . . . 9 class (𝑥𝑑𝑦)
95, 3, 7co 6811 . . . . . . . . 9 class (𝑦𝑑𝑥)
108, 9wceq 1630 . . . . . . . 8 wff (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
11 vp . . . . . . . . 9 setvar 𝑝
1211cv 1629 . . . . . . . 8 class 𝑝
1310, 4, 12wral 3048 . . . . . . 7 wff 𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
1413, 2, 12wral 3048 . . . . . 6 wff 𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
15 vz . . . . . . . . . . . . 13 setvar 𝑧
1615cv 1629 . . . . . . . . . . . 12 class 𝑧
1716, 16, 7co 6811 . . . . . . . . . . 11 class (𝑧𝑑𝑧)
188, 17wceq 1630 . . . . . . . . . 10 wff (𝑥𝑑𝑦) = (𝑧𝑑𝑧)
192, 4weq 2038 . . . . . . . . . 10 wff 𝑥 = 𝑦
2018, 19wi 4 . . . . . . . . 9 wff ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2120, 15, 12wral 3048 . . . . . . . 8 wff 𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2221, 4, 12wral 3048 . . . . . . 7 wff 𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2322, 2, 12wral 3048 . . . . . 6 wff 𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2414, 23wa 383 . . . . 5 wff (∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
25 vf . . . . . . 7 setvar 𝑓
2625cv 1629 . . . . . 6 class 𝑓
27 cds 16150 . . . . . 6 class dist
2826, 27cfv 6047 . . . . 5 class (dist‘𝑓)
2924, 6, 28wsbc 3574 . . . 4 wff [(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
30 cbs 16057 . . . . 5 class Base
3126, 30cfv 6047 . . . 4 class (Base‘𝑓)
3229, 11, 31wsbc 3574 . . 3 wff [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
3332, 25cab 2744 . 2 class {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
341, 33wceq 1630 1 wff TarskiGC = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
Colors of variables: wff setvar class
This definition is referenced by:  istrkgc  25550
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